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Theorem sb3 2477
Description: One direction of a simplified definition of substitution when variables are distinct. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
Assertion
Ref Expression
sb3 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))

Proof of Theorem sb3
StepHypRef Expression
1 sb3b 2476 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
21biimprd 247 1 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wal 1540  wex 1782  [wsb 2068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-12 2172  ax-13 2372
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069
This theorem is referenced by:  dfsb1  2481
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