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Theorem nfsbc 3795
Description: Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2363. Use the weaker nfsbcw 3792 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 1798 . . 3 𝑦
2 nfsbc.1 . . . 4 𝑥𝐴
32a1i 11 . . 3 (⊤ → 𝑥𝐴)
4 nfsbc.2 . . . 4 𝑥𝜑
54a1i 11 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfsbcd 3794 . 2 (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
76mptru 1540 1 𝑥[𝐴 / 𝑦]𝜑
Colors of variables: wff setvar class
Syntax hints:  wtru 1534  wnf 1777  wnfc 2875  [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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2363  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-sbc 3771
This theorem is referenced by:  cbvralcsf  3931  ralrnmpt  7088  elovmporab1  7648  opreu2reuALT  32211
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