df-∃1-alt4

Theorem.

Arguments:

x (sv), y (sv), phi (pr), psi (pr),

Distinct variable conditions:

(x, y), (x, psi), (y, phi),

Hypotheses:

((xeqy)to(phileftrightarrowpsi))

Assertions:

(_e1xphileftrightarrow(existsxphiwedgeforallxforally((phiwedgepsi)to(xeqy))))

Proof:

Hyp Ref Line Expr
Hypo1((xeqy)to(phileftrightarrowpsi))
df-∃1-alt52(_e1xphileftrightarrow(existsxphiwedge_em1xphi))
1df-∃mo-alt43(_em1xphileftrightarrowforallxforally((phiwedgepsi)to(xeqy)))
3bi.bldan>4((existsxphiwedge_em1xphi)leftrightarrow(existsxphiwedgeforallxforally((phiwedgepsi)to(xeqy))))
2, 4bitr5(_e1xphileftrightarrow(existsxphiwedgeforallxforally((phiwedgepsi)to(xeqy))))