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Theorem sbc19.21g 2929
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 2902 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
2 sbcgf.1 . . . 4 𝑥𝜑
32sbcgf 2928 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))
43imbi1d 230 . 2 (𝐴𝑉 → (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ↔ (𝜑[𝐴 / 𝑥]𝜓)))
51, 4bitrd 187 1 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜑[𝐴 / 𝑥]𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1404  wcel 1448  [wsbc 2862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sbc 2863
This theorem is referenced by: (None)
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