Theorem sb4a 1815

Description: A version of sb4 1846 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Assertion

Ref	Expression
sb4a	⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Proof of Theorem sb4a

Step	Hyp	Ref	Expression
1		sb1 1780	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
2		equs5a 1808	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	syl 14	1 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362  ∃wex 1506  [wsb 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-gen 1463  ax-ie2 1508  ax-11 1520  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by:  sb6f  1817  hbsb2a  1820