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Theorem sbcnestgf 3986
Description: Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
sbcnestgf ((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))

Proof of Theorem sbcnestgf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3431 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
2 csbeq1 3529 . . . . . 6 (𝑧 = 𝐴𝑧 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
32sbceq1d 3434 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
41, 3bibi12d 335 . . . 4 (𝑧 = 𝐴 → (([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑) ↔ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑)))
54imbi2d 330 . . 3 (𝑧 = 𝐴 → ((∀𝑦𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑)) ↔ (∀𝑦𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))))
6 vex 3198 . . . . 5 𝑧 ∈ V
76a1i 11 . . . 4 (∀𝑦𝑥𝜑𝑧 ∈ V)
8 csbeq1a 3535 . . . . . 6 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
98sbceq1d 3434 . . . . 5 (𝑥 = 𝑧 → ([𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
109adantl 482 . . . 4 ((∀𝑦𝑥𝜑𝑥 = 𝑧) → ([𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
11 nfnf1 2029 . . . . 5 𝑥𝑥𝜑
1211nfal 2151 . . . 4 𝑥𝑦𝑥𝜑
13 nfa1 2026 . . . . 5 𝑦𝑦𝑥𝜑
14 nfcsb1v 3542 . . . . . 6 𝑥𝑧 / 𝑥𝐵
1514a1i 11 . . . . 5 (∀𝑦𝑥𝜑𝑥𝑧 / 𝑥𝐵)
16 sp 2051 . . . . 5 (∀𝑦𝑥𝜑 → Ⅎ𝑥𝜑)
1713, 15, 16nfsbcd 3450 . . . 4 (∀𝑦𝑥𝜑 → Ⅎ𝑥[𝑧 / 𝑥𝐵 / 𝑦]𝜑)
187, 10, 12, 17sbciedf 3465 . . 3 (∀𝑦𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
195, 18vtoclg 3261 . 2 (𝐴𝑉 → (∀𝑦𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑)))
2019imp 445 1 ((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479   = wceq 1481  wnf 1706  wcel 1988  wnfc 2749  Vcvv 3195  [wsbc 3429  csb 3526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-sbc 3430  df-csb 3527
This theorem is referenced by:  csbnestgf  3987  sbcnestg  3988
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