Theorem 19.36 2232

Description: Theorem 19.36 of [Margaris] p. 90. See 19.36v 1999 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-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 1884	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.36.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	19.9 2207	. . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	3	imbi2i 339	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓))
5	1, 4	bitri 278	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1540  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-12 2179

This theorem depends on definitions:  df-bi 210  df-ex 1787  df-nf 1791

This theorem is referenced by:  19.36i  2233  19.12vv  2350  spcimgft  3491  19.12b  33351