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Theorem sbcco 3795
Description: A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcco ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
Distinct variable group:   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)

Proof of Theorem sbcco
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbcex 3779 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑𝐴 ∈ V)
2 sbcex 3779 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
3 dfsbcq 3771 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑦][𝑦 / 𝑥]𝜑))
4 dfsbcq 3771 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
5 sbsbc 3773 . . . . . 6 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
65sbbii 2072 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
7 nfv 1906 . . . . . 6 𝑦𝜑
87sbco2 2546 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
9 sbsbc 3773 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
106, 8, 93bitr3ri 303 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
11 sbsbc 3773 . . . 4 ([𝑧 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
1210, 11bitri 276 . . 3 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
133, 4, 12vtoclbg 3566 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
141, 2, 13pm5.21nii 380 1 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  [wsb 2060  wcel 2105  Vcvv 3492  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-v 3494  df-sbc 3770
This theorem is referenced by:  csbco  3896  sbccom2  35284  sbccom2f  35285
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