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Theorem sbc19.21g 2949
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 2922 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
2 sbcgf.1 . . . 4 𝑥𝜑
32sbcgf 2948 . . 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 1421  wcel 1465  [wsbc 2882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883
This theorem is referenced by: (None)
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