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Theorem sb3 2492
 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 2488 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
2 sb2 2489 . 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 383  ∀wal 1630  ∃wex 1853  [wsb 2046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-12 2196  ax-13 2391 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-sb 2047 This theorem is referenced by:  bj-sb3b  33132  wl-sb5nae  33671
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