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Theorem sbccow 3780
Description: A composition law for class substitution. Version of sbcco 3783 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 26-Sep-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
Assertion
Ref Expression
sbccow ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)

Proof of Theorem sbccow
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbcex 3767 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑𝐴 ∈ V)
2 sbcex 3767 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
3 dfsbcq 3759 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑦][𝑦 / 𝑥]𝜑))
4 dfsbcq 3759 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
5 sbsbc 3761 . . . . . 6 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
65sbbii 2082 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
7 sbco2vv 2109 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
8 sbsbc 3761 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
96, 7, 83bitr3ri 305 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
10 sbsbc 3761 . . . 4 ([𝑧 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
119, 10bitri 278 . . 3 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
123, 4, 11vtoclbg 3554 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
131, 2, 12pm5.21nii 383 1 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  [wsb 2070  wcel 2115  Vcvv 3479  [wsbc 3757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3481  df-sbc 3758
This theorem is referenced by:  sbc7  3788  sbccom  3837  sbcralt  3838  csbcow  3880  2nreu  4374  bnj62  32010  bnj610  32038  bnj976  32069  bnj1468  32138  aomclem6  39850
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