Theorem spimev 2424

Description: Distinct-variable version of spime 2421. (Contributed by NM, 10-Jan-1993.) Usage of this theorem is discouraged because it depends on ax-13 2404. Use spimevw 2015 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 1944	. 2 ⊢ Ⅎ𝑥𝜑
2		spimev.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spime 2421	1 ⊢ (𝜑 → ∃𝑥𝜓)

This proof depends on syntax axioms:   → wi 4  ∃wex 1809

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-12 2213  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-nf 1814

This theorem is used by: (None)