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Theorem sb4e 2463
Description: One direction of a simplified definition of substitution that unlike sb4 2457 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 2013 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 equs5e 2450 . 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 383  wal 1594  wex 1817  [wsb 2010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-10 2132  ax-12 2160  ax-13 2355
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1818  df-nf 1823  df-sb 2011
This theorem is referenced by:  hbsb2e  2464
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