Theorem 19.23h 1546

Description: Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.)

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	ax-gen 1497	. 2 ⊢ ∀𝑥(𝜓 → ∀𝑥𝜓)
3		19.23ht 1545	. 2 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
4	2, 3	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1395  ∃wex 1540

This theorem was proved from axioms:  ax-mp 5  ax-gen 1497  ax-ie2 1542

This theorem is referenced by:  alnex  1547  19.8a  1638  exlimih  1641  exlimdh  1644  nf2  1716  equs5or  1878  19.23v  1931  pm11.53  1944