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Theorem sb4a 1729
Description: A version of sb4 1760 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
sb4a ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem sb4a
StepHypRef Expression
1 sb1 1696 . 2 ([𝑦 / 𝑥]∀𝑦𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
2 equs5a 1722 . 2 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
31, 2syl 14 1 ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287  wex 1426  [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-gen 1383  ax-ie2 1428  ax-11 1442  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-sb 1693
This theorem is referenced by:  sb6f  1731  hbsb2a  1734
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