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Theorem sb3 2353
 Description: One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sb3 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))

Proof of Theorem sb3
StepHypRef Expression
1 equs5 2349 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
2 sb2 2350 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
31, 2syl6bi 243 1 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1479  ∃wex 1702  [wsb 1878 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045  ax-13 2244 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1703  df-nf 1708  df-sb 1879 This theorem is referenced by:  bj-sb3b  32779  wl-sb5nae  33311
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