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

Theorem nfcsb 3908
Description: Bound-variable hypothesis builder for substitution into a class. Usage of this theorem is discouraged because it depends on ax-13 2383. Use the weaker nfcsbw 3907 when possible. (Contributed by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfcsb.1 𝑥𝐴
nfcsb.2 𝑥𝐵
Assertion
Ref Expression
nfcsb 𝑥𝐴 / 𝑦𝐵

Proof of Theorem nfcsb
StepHypRef Expression
1 nftru 1798 . . 3 𝑦
2 nfcsb.1 . . . 4 𝑥𝐴
32a1i 11 . . 3 (⊤ → 𝑥𝐴)
4 nfcsb.2 . . . 4 𝑥𝐵
54a1i 11 . . 3 (⊤ → 𝑥𝐵)
61, 3, 5nfcsbd 3906 . 2 (⊤ → 𝑥𝐴 / 𝑦𝐵)
76mptru 1537 1 𝑥𝐴 / 𝑦𝐵
Colors of variables: wff setvar class
Syntax hints:  wtru 1531  wnfc 2959  csb 3881
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2383  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-sbc 3771  df-csb 3882
This theorem is referenced by:  cbvralcsf  3923  cbvreucsf  3925  cbvrabcsf  3926  elfvmptrab1  6788  elovmporab1  7385  nfsum  15040
  Copyright terms: Public domain W3C validator