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

Theorem sbcbii 3457
Description: Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
Hypothesis
Ref Expression
sbcbii.1 (𝜑𝜓)
Assertion
Ref Expression
sbcbii ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)

Proof of Theorem sbcbii
StepHypRef Expression
1 sbcbii.1 . . . 4 (𝜑𝜓)
21a1i 11 . . 3 (⊤ → (𝜑𝜓))
32sbcbidv 3456 . 2 (⊤ → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
43trud 1483 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wtru 1475  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-sbc 3402
This theorem is referenced by:  eqsbc3r  3458  eqsbc3rOLD  3459  sbc3an  3460  sbccomlem  3474  sbccom  3475  sbcrext  3477  sbcrextOLD  3478  sbcabel  3482  csbco  3508  sbcnel12g  3936  sbcne12  3937  csbcom  3945  csbnestgf  3947  sbccsb  3955  sbccsb2  3956  csbab  3959  sbcssg  4034  sbcrel  5118  difopab  5163  sbcfung  5813  tfinds2  6932  mpt2xopovel  7210  f1od2  28693  bnj62  29846  bnj89  29847  bnj156  29856  bnj524  29866  bnj538OLD  29870  bnj610  29877  bnj919  29897  bnj976  29908  bnj110  29988  bnj91  29991  bnj92  29992  bnj106  29998  bnj121  30000  bnj124  30001  bnj125  30002  bnj126  30003  bnj130  30004  bnj154  30008  bnj155  30009  bnj153  30010  bnj207  30011  bnj523  30017  bnj526  30018  bnj539  30021  bnj540  30022  bnj581  30038  bnj591  30041  bnj609  30047  bnj611  30048  bnj934  30065  bnj1000  30071  bnj984  30082  bnj985  30083  bnj1040  30100  bnj1123  30114  bnj1452  30180  bnj1463  30183  sbcalf  32883  sbcexf  32884  sbccom2lem  32895  sbccom2  32896  sbccom2f  32897  sbccom2fi  32898  csbcom2fi  32900  2sbcrex  36162  2sbcrexOLD  36164  sbcrot3  36169  sbcrot5  36170  2rexfrabdioph  36174  3rexfrabdioph  36175  4rexfrabdioph  36176  6rexfrabdioph  36177  7rexfrabdioph  36178  rmydioph  36395  expdiophlem2  36403  sbcheg  36889  sbc3or  37555  sbcbiiOLD  37558  trsbc  37567  onfrALTlem5  37574  csbabgOLD  37868
  Copyright terms: Public domain W3C validator