Theorem 19.36 2229

Description: Theorem 19.36 of [Margaris] p. 90. See 19.36v 1986 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 1876	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.36.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	19.9 2204	. . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	3	imbi2i 336	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓))
5	1, 4	bitri 275	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537  ∃wex 1778  Ⅎwnf 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-ex 1779  df-nf 1783

This theorem is referenced by:  19.36i  2230  19.12vv  2347  spcimgfi1OLD  3531  19.12b  35761