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Theorem nfsbc 3741
Description: Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfsbcw 3738 when possible. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfsbc.1 𝑥𝐴
nfsbc.2 𝑥𝜑
Assertion
Ref Expression
nfsbc 𝑥[𝐴 / 𝑦]𝜑

Proof of Theorem nfsbc
StepHypRef Expression
1 nftru 1807 . . 3 𝑦
2 nfsbc.1 . . . 4 𝑥𝐴
32a1i 11 . . 3 (⊤ → 𝑥𝐴)
4 nfsbc.2 . . . 4 𝑥𝜑
54a1i 11 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfsbcd 3740 . 2 (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
76mptru 1546 1 𝑥[𝐴 / 𝑦]𝜑
Colors of variables: wff setvar class
Syntax hints:  wtru 1540  wnf 1786  wnfc 2887  [wsbc 3716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-sbc 3717
This theorem is referenced by:  cbvralcsf  3877  ralrnmpt  6972  elovmporab1  7517  opreu2reuALT  30825
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