Theorem spimed 1738

Description: Deduction version of spime 1739. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)

Hypotheses

Ref	Expression
spimed.1	⊢ (𝜒 → Ⅎ𝑥𝜑)
spimed.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimed	⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))

Proof of Theorem spimed

Step	Hyp	Ref	Expression
1		spimed.1	. . 3 ⊢ (𝜒 → Ⅎ𝑥𝜑)
2	1	nfrd 1518	. 2 ⊢ (𝜒 → (𝜑 → ∀𝑥𝜑))
3		a9e 1694	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
4		spimed.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	3, 4	eximii 1600	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
6	5	19.35i 1623	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
7	2, 6	syl6 33	1 ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1351  Ⅎwnf 1458  ∃wex 1490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-4 1508  ax-i9 1528  ax-ial 1532

This theorem depends on definitions:  df-bi 117  df-nf 1459

This theorem is referenced by:  spime  1739