Theorem 19.23 2210

Description: Theorem 19.23 of [Margaris] p. 90. See 19.23v 1942 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 2209	. 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-ex 1780  df-nf 1784

This theorem is referenced by:  exlimi  2216  equsalv  2267  nf5  2289  19.23h  2295  pm11.53  2366  equsal  2438  2sb6rf  2496  r19.3rz  4445  ralidm  4458  ssrelf  30369  bj-biexal1  34043  bj-biexex  34047  axc11n-16  36078  axc11next  40744