Theorem spimedv 2197

Description: Deduction version of spimev 2410. Version of spimed 2406 with a disjoint variable condition, which does not require ax-13 2390. See spime 2407 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 2196	. 2 ⊢ (𝜒 → (𝜑 → ∀𝑥𝜑))
3		ax6ev 1972	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
4		spimedv.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	3, 4	eximii 1837	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
6	5	19.35i 1879	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
7	2, 6	syl6 35	1 ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-nf 1785

This theorem is referenced by:  spimefv  2198