Theorem 19.36 1871

Description: Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
19.36.1	⊢ Ⅎxψ

Assertion

Ref	Expression
19.36	⊢ (∃x(φ → ψ) ↔ (∀xφ → ψ))

Proof of Theorem 19.36

Step	Hyp	Ref	Expression
1		19.35 1600	. 2 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ∃xψ))
2		19.36.1	. . . 4 ⊢ Ⅎxψ
3	2	19.9 1783	. . 3 ⊢ (∃xψ ↔ ψ)
4	3	imbi2i 303	. 2 ⊢ ((∀xφ → ∃xψ) ↔ (∀xφ → ψ))
5	1, 4	bitri 240	1 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  19.36i  1872  19.36v  1896  19.12vv  1898  spcimgft  2931