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Theorem sbcco 3425
 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 3412 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑𝐴 ∈ V)
2 sbcex 3412 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
3 dfsbcq 3404 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑦][𝑦 / 𝑥]𝜑))
4 dfsbcq 3404 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
5 sbsbc 3406 . . . . . 6 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
65sbbii 1874 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
7 nfv 1830 . . . . . 6 𝑦𝜑
87sbco2 2403 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
9 sbsbc 3406 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝑧 / 𝑦][𝑦 / 𝑥]𝜑)
106, 8, 93bitr3ri 290 . . . 4 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
11 sbsbc 3406 . . . 4 ([𝑧 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
1210, 11bitri 263 . . 3 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑[𝑧 / 𝑥]𝜑)
133, 4, 12vtoclbg 3240 . 2 (𝐴 ∈ V → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
141, 2, 13pm5.21nii 367 1 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  [wsb 1867   ∈ wcel 1977  Vcvv 3173  [wsbc 3402 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-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by:  sbc7  3430  sbccom  3476  sbcralt  3477  csbco  3509  bnj62  30040  bnj610  30071  bnj976  30102  bnj1468  30170  sbccom2  33100  sbccom2f  33101  aomclem6  36647
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