Theorem speiv 1855

Description: Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)

Hypotheses

Ref	Expression
speiv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
speiv.2	⊢ 𝜓

Assertion

Ref	Expression
speiv	⊢ ∃𝑥𝜑

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem speiv

Step	Hyp	Ref	Expression
1		speiv.2	. 2 ⊢ 𝜓
2		speiv.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	biimprd 157	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
4	3	spimev 1854	. 2 ⊢ (𝜓 → ∃𝑥𝜑)
5	1, 4	ax-mp 5	1 ⊢ ∃𝑥𝜑

Syntax hints:   → wi 4   ↔ wb 104  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454

This theorem is referenced by: (None)