Theorem spimev 2295

Description: Distinct-variable version of spime 2292. (Contributed by NM, 10-Jan-1993.)

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 1883	. 2 ⊢ Ⅎ𝑥𝜑
2		spimev.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spime 2292	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750

This theorem is referenced by:  axsep  4813  dtru  4887  zfpair  4934  fvn0ssdmfun  6390  refimssco  38230  rlimdmafv  41578  elsprel  42050