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Theorem sb4e 1733
Description: One direction of a simplified definition of substitution that unlike sb4 1760 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
sb4e ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem sb4e
StepHypRef Expression
1 sb1 1696 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 equs5e 1723 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
31, 2syl 14 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃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-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-11 1442  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-sb 1693
This theorem is referenced by:  hbsb2e  1735
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