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Theorem sbcth2 3845
 Description: A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
sbcth2.1 (𝑥𝐵𝜑)
Assertion
Ref Expression
sbcth2 (𝐴𝐵[𝐴 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem sbcth2
StepHypRef Expression
1 sbcth2.1 . . 3 (𝑥𝐵𝜑)
21rgen 3135 . 2 𝑥𝐵 𝜑
3 rspsbc 3840 . 2 (𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))
42, 3mpi 20 1 (𝐴𝐵[𝐴 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3125  [wsbc 3752 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-sbc 3753 This theorem is referenced by: (None)
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