Theorem spimed 2422

Description: Deduction version of spime 2423. (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 2406. Use spimedv 2236 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 2235	. 2 ⊢ (𝜒 → (𝜑 → ∀𝑥𝜑))
3		ax6e 2417	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
4		spimed.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	3, 4	eximii 1870	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
6	5	19.35i 1911	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
7	2, 6	syl6 36	1 ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  spime  2423  2ax6elem  2504