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Theorem sbcnestgf 4418
Description: Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker sbcnestgfw 4413 when possible. (Contributed by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)
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 3774 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
2 csbeq1 3891 . . . . . 6 (𝑧 = 𝐴𝑧 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
32sbceq1d 3777 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
41, 3bibi12d 345 . . . 4 (𝑧 = 𝐴 → (([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑) ↔ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑)))
54imbi2d 340 . . 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 3472 . . . . 5 𝑧 ∈ V
76a1i 11 . . . 4 (∀𝑦𝑥𝜑𝑧 ∈ V)
8 csbeq1a 3902 . . . . . 6 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
98sbceq1d 3777 . . . . 5 (𝑥 = 𝑧 → ([𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
109adantl 481 . . . 4 ((∀𝑦𝑥𝜑𝑥 = 𝑧) → ([𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
11 nfnf1 2143 . . . . 5 𝑥𝑥𝜑
1211nfal 2310 . . . 4 𝑥𝑦𝑥𝜑
13 nfa1 2140 . . . . 5 𝑦𝑦𝑥𝜑
14 nfcsb1v 3913 . . . . . 6 𝑥𝑧 / 𝑥𝐵
1514a1i 11 . . . . 5 (∀𝑦𝑥𝜑𝑥𝑧 / 𝑥𝐵)
16 sp 2168 . . . . 5 (∀𝑦𝑥𝜑 → Ⅎ𝑥𝜑)
1713, 15, 16nfsbcd 3796 . . . 4 (∀𝑦𝑥𝜑 → Ⅎ𝑥[𝑧 / 𝑥𝐵 / 𝑦]𝜑)
187, 10, 12, 17sbciedf 3816 . . 3 (∀𝑦𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
195, 18vtoclg 3537 . 2 (𝐴𝑉 → (∀𝑦𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑)))
2019imp 406 1 ((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531   = wceq 1533  wnf 1777  wcel 2098  wnfc 2877  Vcvv 3468  [wsbc 3772  csb 3888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-v 3470  df-sbc 3773  df-csb 3889
This theorem is referenced by:  csbnestgf  4419  sbcnestg  4420
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