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Theorem sbcnestgf 4338
Description: Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker sbcnestgfw 4333 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 3696 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
2 csbeq1 3814 . . . . . 6 (𝑧 = 𝐴𝑧 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
32sbceq1d 3699 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
41, 3bibi12d 349 . . . 4 (𝑧 = 𝐴 → (([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑) ↔ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑)))
54imbi2d 344 . . 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 3412 . . . . 5 𝑧 ∈ V
76a1i 11 . . . 4 (∀𝑦𝑥𝜑𝑧 ∈ V)
8 csbeq1a 3825 . . . . . 6 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
98sbceq1d 3699 . . . . 5 (𝑥 = 𝑧 → ([𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
109adantl 485 . . . 4 ((∀𝑦𝑥𝜑𝑥 = 𝑧) → ([𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
11 nfnf1 2155 . . . . 5 𝑥𝑥𝜑
1211nfal 2322 . . . 4 𝑥𝑦𝑥𝜑
13 nfa1 2152 . . . . 5 𝑦𝑦𝑥𝜑
14 nfcsb1v 3836 . . . . . 6 𝑥𝑧 / 𝑥𝐵
1514a1i 11 . . . . 5 (∀𝑦𝑥𝜑𝑥𝑧 / 𝑥𝐵)
16 sp 2180 . . . . 5 (∀𝑦𝑥𝜑 → Ⅎ𝑥𝜑)
1713, 15, 16nfsbcd 3718 . . . 4 (∀𝑦𝑥𝜑 → Ⅎ𝑥[𝑧 / 𝑥𝐵 / 𝑦]𝜑)
187, 10, 12, 17sbciedf 3738 . . 3 (∀𝑦𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑[𝑧 / 𝑥𝐵 / 𝑦]𝜑))
195, 18vtoclg 3481 . 2 (𝐴𝑉 → (∀𝑦𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑)))
2019imp 410 1 ((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1541   = wceq 1543  wnf 1791  wcel 2110  wnfc 2884  Vcvv 3408  [wsbc 3694  csb 3811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-v 3410  df-sbc 3695  df-csb 3812
This theorem is referenced by:  csbnestgf  4339  sbcnestg  4340
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