Theorem 19.23 2178

Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1924 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 2177	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1523  ∃wex 1765  Ⅎwnf 1769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-12 2143

This theorem depends on definitions:  df-bi 208  df-ex 1766  df-nf 1770

This theorem is referenced by:  exlimi  2184  equsalv  2233  nf5  2258  19.23h  2264  pm11.53  2325  equsal  2397  2sb6rf  2456  2sb6rfOLD  2457  r19.3rz  4362  ralidm  4375  ssrelf  30052  bj-biexal1  33643  bj-biexex  33647  axc11n-16  35626  axc11next  40297