Theorem spimedv 2232

Description: Deduction version of spimev 2423. Version of spimed 2419 with a disjoint variable condition, which does not require ax-13 2403. See spime 2420 for a non-deduction version. (Contributed by NM, 14-May-1993.) (Revised by BJ, 31-May-2019.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦)   𝜒(𝑥, 𝑦)

Proof of Theorem spimedv

Step	Hyp	Ref	Expression
1		spimedv.1	. . 3 ⊢ (𝜒 → Ⅎ𝑥𝜑)
2	1	nf5rd 2231	. 2 ⊢ (𝜒 → (𝜑 → ∀𝑥𝜑))
3		ax6ev 1998	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
4		spimedv.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	3, 4	eximii 1866	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
6	5	19.35i 1907	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
7	2, 6	syl6 36	1 ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1808  Ⅎwnf 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1809  df-nf 1813

This theorem is used by:  spimefv  2233