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Theorem csbcom 3946
 Description: Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
csbcom 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem csbcom
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbccom 3476 . . . 4 ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐵 / 𝑦][𝐴 / 𝑥]𝑧𝐶)
2 sbcel2 3941 . . . . 5 ([𝐵 / 𝑦]𝑧𝐶𝑧𝐵 / 𝑦𝐶)
32sbcbii 3458 . . . 4 ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶)
4 sbcel2 3941 . . . . 5 ([𝐴 / 𝑥]𝑧𝐶𝑧𝐴 / 𝑥𝐶)
54sbcbii 3458 . . . 4 ([𝐵 / 𝑦][𝐴 / 𝑥]𝑧𝐶[𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶)
61, 3, 53bitr3i 289 . . 3 ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶)
7 sbcel2 3941 . . 3 ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶𝑧𝐴 / 𝑥𝐵 / 𝑦𝐶)
8 sbcel2 3941 . . 3 ([𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶𝑧𝐵 / 𝑦𝐴 / 𝑥𝐶)
96, 7, 83bitr3i 289 . 2 (𝑧𝐴 / 𝑥𝐵 / 𝑦𝐶𝑧𝐵 / 𝑦𝐴 / 𝑥𝐶)
109eqriv 2607 1 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  [wsbc 3402  ⦋csb 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-nul 3875 This theorem is referenced by:  ovmpt2s  6682  fvmpt2curryd  7284
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