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Theorem sbcth2 3747
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 3131 . 2 𝑥𝐵 𝜑
3 rspsbc 3742 . 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 2164  wral 3117  [wsbc 3662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-v 3416  df-sbc 3663
This theorem is referenced by: (None)
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