Users' Mathboxes Mathbox for Giovanni Mascellani < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbccom2fi Structured version   Visualization version   GIF version

Theorem sbccom2fi 33932
Description: Commutative law for double class substitution, with non free variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
Hypotheses
Ref Expression
sbccom2fi.1 𝐴 ∈ V
sbccom2fi.2 𝑦𝐴
sbccom2fi.3 𝐴 / 𝑥𝐵 = 𝐶
sbccom2fi.4 ([𝐴 / 𝑥]𝜑𝜓)
Assertion
Ref Expression
sbccom2fi ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem sbccom2fi
StepHypRef Expression
1 sbccom2fi.1 . . 3 𝐴 ∈ V
2 sbccom2fi.2 . . 3 𝑦𝐴
31, 2sbccom2f 33931 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
4 sbccom2fi.3 . . 3 𝐴 / 𝑥𝐵 = 𝐶
5 dfsbcq 3437 . . 3 (𝐴 / 𝑥𝐵 = 𝐶 → ([𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑))
64, 5ax-mp 5 . 2 ([𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)
7 sbccom2fi.4 . . 3 ([𝐴 / 𝑥]𝜑𝜓)
87sbcbii 3491 . 2 ([𝐶 / 𝑦][𝐴 / 𝑥]𝜑[𝐶 / 𝑦]𝜓)
93, 6, 83bitri 286 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1483  wcel 1990  wnfc 2751  Vcvv 3200  [wsbc 3435  csb 3533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534
This theorem is referenced by:  csbcom2fi  33934
  Copyright terms: Public domain W3C validator