Theorem spimev 1885

Description: Distinct-variable version of spime 1765. (Contributed by NM, 5-Aug-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 1552	. 2 ⊢ Ⅎ𝑥𝜑
2		spimev.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spime 1765	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485

This theorem is referenced by:  speiv  1886  cbvexvw  1945  rnxpid  5136