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Theorem csbcom2fi 34247
Description: Commutative law for double class substitution in a class, with non free variable condition and in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)
Hypotheses
Ref Expression
csbcom2fi.1 𝐴 ∈ V
csbcom2fi.2 𝑦𝐴
csbcom2fi.3 𝐴 / 𝑥𝐵 = 𝐶
csbcom2fi.4 𝐴 / 𝑥𝐷 = 𝐸
Assertion
Ref Expression
csbcom2fi 𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐸
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)

Proof of Theorem csbcom2fi
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3675 . . . . 5 𝐴 / 𝑥𝐵 / 𝑦𝐷 = {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐷}
21abeq2i 2873 . . . 4 (𝑧𝐴 / 𝑥𝐵 / 𝑦𝐷[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐷)
3 df-csb 3675 . . . . . 6 𝐵 / 𝑦𝐷 = {𝑧[𝐵 / 𝑦]𝑧𝐷}
43abeq2i 2873 . . . . 5 (𝑧𝐵 / 𝑦𝐷[𝐵 / 𝑦]𝑧𝐷)
54sbcbii 3632 . . . 4 ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐷[𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐷)
62, 5bitri 264 . . 3 (𝑧𝐴 / 𝑥𝐵 / 𝑦𝐷[𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐷)
7 csbcom2fi.1 . . . 4 𝐴 ∈ V
8 csbcom2fi.2 . . . 4 𝑦𝐴
9 csbcom2fi.3 . . . 4 𝐴 / 𝑥𝐵 = 𝐶
10 df-csb 3675 . . . . . 6 𝐴 / 𝑥𝐷 = {𝑧[𝐴 / 𝑥]𝑧𝐷}
1110abeq2i 2873 . . . . 5 (𝑧𝐴 / 𝑥𝐷[𝐴 / 𝑥]𝑧𝐷)
12 csbcom2fi.4 . . . . . 6 𝐴 / 𝑥𝐷 = 𝐸
1312eleq2i 2831 . . . . 5 (𝑧𝐴 / 𝑥𝐷𝑧𝐸)
1411, 13bitr3i 266 . . . 4 ([𝐴 / 𝑥]𝑧𝐷𝑧𝐸)
157, 8, 9, 14sbccom2fi 34245 . . 3 ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐷[𝐶 / 𝑦]𝑧𝐸)
16 sbcel2 4132 . . 3 ([𝐶 / 𝑦]𝑧𝐸𝑧𝐶 / 𝑦𝐸)
176, 15, 163bitri 286 . 2 (𝑧𝐴 / 𝑥𝐵 / 𝑦𝐷𝑧𝐶 / 𝑦𝐸)
1817eqriv 2757 1 𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐸
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  wnfc 2889  Vcvv 3340  [wsbc 3576  csb 3674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-nul 4059
This theorem is referenced by: (None)
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