Theorem 19.23 2219

Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1944 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 2218	. 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 1540  ∃wex 1781  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-ex 1782  df-nf 1786

This theorem is referenced by:  exlimi  2225  equsalv  2275  nf5  2289  19.23h  2295  pm11.53  2351  equsal  2422  2sb6rf  2478  ceqsal  3480  r19.3rz  4456  ssrelf  32704  bj-biexal1  36947  bj-biexex  36951  axc11n-16  39311  axc11next  44759  r19.3rzf  45514