Theorem spimev 2386

Description: Distinct-variable version of spime 2383. (Contributed by NM, 10-Jan-1993.) Usage of this theorem is discouraged because it depends on ax-13 2366. Use spimevw 1991 instead. (New usage is discouraged.)

Hypothesis

Ref	Expression
spimev.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimev	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem spimev

Step	Hyp	Ref	Expression
1		nfv 1910	. 2 ⊢ Ⅎ𝑥𝜑
2		spimev.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spime 2383	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1774

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 2164  ax-13 2366

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-nf 1779

This theorem is referenced by: (None)