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Theorem sbcco 3685
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 3672 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑𝐴 ∈ V)
2 sbcex 3672 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
3 dfsbcq 3664 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑦][𝑦 / 𝑥]𝜑))
4 dfsbcq 3664 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
5 sbsbc 3666 . . . . . 6 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
65sbbii 2074 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
7 nfv 2013 . . . . . 6 𝑦𝜑
87sbco2 2547 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
9 sbsbc 3666 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
106, 8, 93bitr3ri 294 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
11 sbsbc 3666 . . . 4 ([𝑧 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
1210, 11bitri 267 . . 3 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
133, 4, 12vtoclbg 3483 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
141, 2, 13pm5.21nii 370 1 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 198  [wsb 2067  wcel 2164  Vcvv 3414  [wsbc 3662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416  df-sbc 3663
This theorem is referenced by:  sbc7  3690  sbccom  3734  sbcralt  3735  csbco  3767  bnj62  31324  bnj610  31352  bnj976  31383  bnj1468  31451  sbccom2  34463  sbccom2f  34464  aomclem6  38465
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