Theorem 19.23 2248

Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1975 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 2247	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  exlimi  2254  equsalv  2302  nf5  2316  19.23h  2322  pm11.53  2376  equsal  2447  2sb6rf  2503  ceqsal  3488  r19.3rz  4457  ssrelf  33202  bj-biexal1  37587  bj-biexex  37591  axc11n-16  39975  axc11next  45375  r19.3rzf  46142