Theorem spimed 2383

Description: Deduction version of spime 2384. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) Usage of this theorem is discouraged because it depends on ax-13 2367. Use spimedv 2186 instead. (New usage is discouraged.)

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	nf5rd 2185	. 2 ⊢ (𝜒 → (𝜑 → ∀𝑥𝜑))
3		ax6e 2378	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
4		spimed.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	3, 4	eximii 1832	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
6	5	19.35i 1874	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
7	2, 6	syl6 35	1 ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1532  ∃wex 1774  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-12 2167  ax-13 2367

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779

This theorem is referenced by:  spime  2384  2ax6elem  2465