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

Theorem sbccom2fi 34062
 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 34061 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
4 sbccom2fi.3 . . 3 𝐴 / 𝑥𝐵 = 𝐶
5 dfsbcq 3470 . . 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 3524 . 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 1523   ∈ wcel 2030  Ⅎwnfc 2780  Vcvv 3231  [wsbc 3468  ⦋csb 3566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469  df-csb 3567 This theorem is referenced by:  csbcom2fi  34064
 Copyright terms: Public domain W3C validator