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Theorem csbcom 4412
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 3860 . . . 4 ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐵 / 𝑦][𝐴 / 𝑥]𝑧𝐶)
2 sbcel2 4410 . . . . 5 ([𝐵 / 𝑦]𝑧𝐶𝑧𝐵 / 𝑦𝐶)
32sbcbii 3832 . . . 4 ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶)
4 sbcel2 4410 . . . . 5 ([𝐴 / 𝑥]𝑧𝐶𝑧𝐴 / 𝑥𝐶)
54sbcbii 3832 . . . 4 ([𝐵 / 𝑦][𝐴 / 𝑥]𝑧𝐶[𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶)
61, 3, 53bitr3i 301 . . 3 ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶)
7 sbcel2 4410 . . 3 ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶𝑧𝐴 / 𝑥𝐵 / 𝑦𝐶)
8 sbcel2 4410 . . 3 ([𝐵 / 𝑦]𝑧𝐴 / 𝑥𝐶𝑧𝐵 / 𝑦𝐴 / 𝑥𝐶)
96, 7, 83bitr3i 301 . 2 (𝑧𝐴 / 𝑥𝐵 / 𝑦𝐶𝑧𝐵 / 𝑦𝐴 / 𝑥𝐶)
109eqriv 2723 1 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  [wsbc 3772  csb 3888
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-nul 4318
This theorem is referenced by:  ovmpos  7551  fvmpocurryd  8254  f1od2  32451
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