Theorem 19.23 2118

Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1911 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 2117	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521  ∃wex 1744  Ⅎwnf 1748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087

This theorem depends on definitions:  df-bi 197  df-or 384  df-ex 1745  df-nf 1750

This theorem is referenced by:  exlimi  2124  equsalv  2146  nf5  2154  19.23h  2160  pm11.53  2215  equsal  2327  2sb6rf  2480  r19.3rz  4095  ralidm  4108  ssrelf  29553  bj-biexal1  32821  bj-biexex  32825  axc11n-16  34542  axc11next  38924