Theorem spimev 2400

Description: Distinct-variable version of spime 2397. (Contributed by NM, 10-Jan-1993.) Usage of this theorem is discouraged because it depends on ax-13 2380. Use spimevw 1992 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 1921	. 2 ⊢ Ⅎ𝑥𝜑
2		spimev.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spime 2397	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-nf 1791

This theorem is referenced by: (None)