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Theorem sbc19.21g 3496
Description: Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
Hypothesis
Ref Expression
sbcgf.1 𝑥𝜑
Assertion
Ref Expression
sbc19.21g (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜑[𝐴 / 𝑥]𝜓)))

Proof of Theorem sbc19.21g
StepHypRef Expression
1 sbcimg 3471 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
2 sbcgf.1 . . . 4 𝑥𝜑
32sbcgf 3495 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))
43imbi1d 331 . 2 (𝐴𝑉 → (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ↔ (𝜑[𝐴 / 𝑥]𝜓)))
51, 4bitrd 268 1 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜑[𝐴 / 𝑥]𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wnf 1706  wcel 1988  [wsbc 3429
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-9 1997  ax-10 2017  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-v 3197  df-sbc 3430
This theorem is referenced by:  bnj121  30914  bnj124  30915  bnj130  30918  bnj207  30925  bnj611  30962  bnj1000  30985
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