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Theorem sbcralt 3805
Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
Assertion
Ref Expression
sbcralt ((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
Distinct variable groups:   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcralt
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbccow 3739 . 2 ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑)
2 simpl 483 . . 3 ((𝐴𝑉𝑦𝐴) → 𝐴𝑉)
3 sbsbc 3720 . . . . 5 ([𝑧 / 𝑥]∀𝑦𝐵 𝜑[𝑧 / 𝑥]𝑦𝐵 𝜑)
4 nfcv 2907 . . . . . . 7 𝑥𝐵
5 nfs1v 2153 . . . . . . 7 𝑥[𝑧 / 𝑥]𝜑
64, 5nfralw 3151 . . . . . 6 𝑥𝑦𝐵 [𝑧 / 𝑥]𝜑
7 sbequ12 2244 . . . . . . 7 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
87ralbidv 3112 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑))
96, 8sbiev 2309 . . . . 5 ([𝑧 / 𝑥]∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑)
103, 9bitr3i 276 . . . 4 ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑)
11 nfnfc1 2910 . . . . . . 7 𝑦𝑦𝐴
12 nfcvd 2908 . . . . . . . 8 (𝑦𝐴𝑦𝑧)
13 id 22 . . . . . . . 8 (𝑦𝐴𝑦𝐴)
1412, 13nfeqd 2917 . . . . . . 7 (𝑦𝐴 → Ⅎ𝑦 𝑧 = 𝐴)
1511, 14nfan1 2193 . . . . . 6 𝑦(𝑦𝐴𝑧 = 𝐴)
16 dfsbcq2 3719 . . . . . . 7 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
1716adantl 482 . . . . . 6 ((𝑦𝐴𝑧 = 𝐴) → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
1815, 17ralbid 3161 . . . . 5 ((𝑦𝐴𝑧 = 𝐴) → (∀𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
1918adantll 711 . . . 4 (((𝐴𝑉𝑦𝐴) ∧ 𝑧 = 𝐴) → (∀𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
2010, 19bitrid 282 . . 3 (((𝐴𝑉𝑦𝐴) ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
212, 20sbcied 3761 . 2 ((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
221, 21bitr3id 285 1 ((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  [wsb 2067  wcel 2106  wnfc 2887  wral 3064  [wsbc 3716
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-v 3434  df-sbc 3717
This theorem is referenced by:  sbcrext  3806  sbcralg  3807
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