Theorem 19.23 2208

Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1939 for a version requiring fewer axioms. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Hypothesis

Ref	Expression
19.23.1	⊢ Ⅎ𝑥𝜓

Assertion

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

Proof of Theorem 19.23

Step	Hyp	Ref	Expression
1		19.23.1	. 2 ⊢ Ⅎ𝑥𝜓
2		19.23t 2207	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1534  ∃wex 1775  Ⅎwnf 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-12 2174

This theorem depends on definitions:  df-bi 207  df-ex 1776  df-nf 1780

This theorem is referenced by:  exlimi  2214  equsalv  2264  nf5  2280  19.23h  2286  pm11.53  2346  equsal  2419  2sb6rf  2475  ceqsal  3516  r19.3rz  4502  ssrelf  32634  bj-biexal1  36687  bj-biexex  36691  axc11n-16  38919  axc11next  44401  r19.3rzf  45100