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Theorem csbcom2fi 33104
 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 3500 . . . . 5 𝐴 / 𝑥𝐵 / 𝑦𝐷 = {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐷}
21abeq2i 2722 . . . 4 (𝑧𝐴 / 𝑥𝐵 / 𝑦𝐷[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐷)
3 df-csb 3500 . . . . . 6 𝐵 / 𝑦𝐷 = {𝑧[𝐵 / 𝑦]𝑧𝐷}
43abeq2i 2722 . . . . 5 (𝑧𝐵 / 𝑦𝐷[𝐵 / 𝑦]𝑧𝐷)
54sbcbii 3458 . . . 4 ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐷[𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐷)
62, 5bitri 263 . . 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 3500 . . . . . 6 𝐴 / 𝑥𝐷 = {𝑧[𝐴 / 𝑥]𝑧𝐷}
1110abeq2i 2722 . . . . 5 (𝑧𝐴 / 𝑥𝐷[𝐴 / 𝑥]𝑧𝐷)
12 csbcom2fi.4 . . . . . 6 𝐴 / 𝑥𝐷 = 𝐸
1312eleq2i 2680 . . . . 5 (𝑧𝐴 / 𝑥𝐷𝑧𝐸)
1411, 13bitr3i 265 . . . 4 ([𝐴 / 𝑥]𝑧𝐷𝑧𝐸)
157, 8, 9, 14sbccom2fi 33102 . . 3 ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐷[𝐶 / 𝑦]𝑧𝐸)
16 sbcel2 3941 . . 3 ([𝐶 / 𝑦]𝑧𝐸𝑧𝐶 / 𝑦𝐸)
176, 15, 163bitri 285 . 2 (𝑧𝐴 / 𝑥𝐵 / 𝑦𝐷𝑧𝐶 / 𝑦𝐸)
1817eqriv 2607 1 𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐸
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  Vcvv 3173  [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-3an 1033  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: (None)
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