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Theorem sbc19.21g 3043
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 3016 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
2 sbcgf.1 . . . 4 𝑥𝜑
32sbcgf 3042 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))
43imbi1d 231 . 2 (𝐴𝑉 → (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ↔ (𝜑[𝐴 / 𝑥]𝜓)))
51, 4bitrd 188 1 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜑[𝐴 / 𝑥]𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wnf 1470  wcel 2158  [wsbc 2974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sbc 2975
This theorem is referenced by: (None)
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