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Theorem sbc19.21g 3844
 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 3818 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
2 sbcgf.1 . . . 4 𝑥𝜑
32sbcgf 3843 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))
43imbi1d 344 . 2 (𝐴𝑉 → (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) ↔ (𝜑[𝐴 / 𝑥]𝜓)))
51, 4bitrd 281 1 (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜑[𝐴 / 𝑥]𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  Ⅎwnf 1777   ∈ wcel 2107  [wsbc 3770 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-sbc 3771 This theorem is referenced by:  bnj121  32130  bnj124  32131  bnj130  32134  bnj207  32141  bnj611  32178  bnj1000  32201
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