MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsbc Structured version   Visualization version   GIF version

Theorem nfsbc 3710
Description: Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker nfsbcw 3707 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 1811 . . 3 𝑦
2 nfsbc.1 . . . 4 𝑥𝐴
32a1i 11 . . 3 (⊤ → 𝑥𝐴)
4 nfsbc.2 . . . 4 𝑥𝜑
54a1i 11 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfsbcd 3709 . 2 (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
76mptru 1549 1 𝑥[𝐴 / 𝑦]𝜑
Colors of variables: wff setvar class
Syntax hints:  wtru 1543  wnf 1790  wnfc 2880  [wsbc 3685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-13 2373  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-sbc 3686
This theorem is referenced by:  cbvralcsf  3842  ralrnmpt  6885  elovmporab1  7422  opreu2reuALT  30412
  Copyright terms: Public domain W3C validator