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Theorem sb4e 2476
Description: One direction of a simplified definition of substitution that unlike sb4b 2466 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2363. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
Assertion
Ref Expression
sb4e ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem sb4e
StepHypRef Expression
1 sb1 2469 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 equs5e 2449 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
31, 2syl 17 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1531  wex 1773  [wsb 2059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163  ax-13 2363
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060
This theorem is referenced by:  hbsb2e  2477
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