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Theorem sbcco 3804
Description: A composition law for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker sbccow 3801 when possible. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) (New usage is discouraged.)
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 3788 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑𝐴 ∈ V)
2 sbcex 3788 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
3 dfsbcq 3780 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑦][𝑦 / 𝑥]𝜑))
4 dfsbcq 3780 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
5 sbsbc 3782 . . . . . 6 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
65sbbii 2080 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
7 nfv 1918 . . . . . 6 𝑦𝜑
87sbco2 2511 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
9 sbsbc 3782 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
106, 8, 93bitr3ri 302 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
11 sbsbc 3782 . . . 4 ([𝑧 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
1210, 11bitri 275 . . 3 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
133, 4, 12vtoclbg 3560 . 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 205  [wsb 2068  wcel 2107  Vcvv 3475  [wsbc 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2372  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-sbc 3779
This theorem is referenced by:  csbco  3910
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