Theorem spsbe 1830

Description: A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.)

Assertion

Ref	Expression
spsbe	⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem spsbe

Step	Hyp	Ref	Expression
1		sb1 1754	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		simpr 109	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑)
3	2	eximi 1588	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥𝜑)
4	1, 3	syl 14	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1480  [wsb 1750

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-sb 1751

This theorem is referenced by:  sbft  1836