Theorem 19.23h 2108

Description: Theorem 19.23 of [Margaris] p. 90. See 19.23 2067. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)

Hypothesis

Ref	Expression
19.23h.1	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

Ref	Expression
19.23h	⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Proof of Theorem 19.23h

Step	Hyp	Ref	Expression
1		19.23h.1	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	nf5i 2011	. 2 ⊢ Ⅎ𝑥𝜓
3	2	19.23 2067	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701

This theorem is referenced by:  equsalhw  2109