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Theorem sbcralt 2862
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 sbcco 2808 . 2 ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑)
2 simpl 106 . . 3 ((𝐴𝑉𝑦𝐴) → 𝐴𝑉)
3 sbsbc 2791 . . . . 5 ([𝑧 / 𝑥]∀𝑦𝐵 𝜑[𝑧 / 𝑥]𝑦𝐵 𝜑)
4 nfcv 2194 . . . . . . 7 𝑥𝐵
5 nfs1v 1831 . . . . . . 7 𝑥[𝑧 / 𝑥]𝜑
64, 5nfralxy 2377 . . . . . 6 𝑥𝑦𝐵 [𝑧 / 𝑥]𝜑
7 sbequ12 1670 . . . . . . 7 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
87ralbidv 2343 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑))
96, 8sbie 1690 . . . . 5 ([𝑧 / 𝑥]∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑)
103, 9bitr3i 179 . . . 4 ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑)
11 nfnfc1 2197 . . . . . . 7 𝑦𝑦𝐴
12 nfcvd 2195 . . . . . . . 8 (𝑦𝐴𝑦𝑧)
13 id 19 . . . . . . . 8 (𝑦𝐴𝑦𝐴)
1412, 13nfeqd 2208 . . . . . . 7 (𝑦𝐴 → Ⅎ𝑦 𝑧 = 𝐴)
1511, 14nfan1 1472 . . . . . 6 𝑦(𝑦𝐴𝑧 = 𝐴)
16 dfsbcq2 2790 . . . . . . 7 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
1716adantl 266 . . . . . 6 ((𝑦𝐴𝑧 = 𝐴) → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
1815, 17ralbid 2341 . . . . 5 ((𝑦𝐴𝑧 = 𝐴) → (∀𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
1918adantll 453 . . . 4 (((𝐴𝑉𝑦𝐴) ∧ 𝑧 = 𝐴) → (∀𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
2010, 19syl5bb 185 . . 3 (((𝐴𝑉𝑦𝐴) ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
212, 20sbcied 2822 . 2 ((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
221, 21syl5bbr 187 1 ((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  [wsb 1661  wnfc 2181  wral 2323  [wsbc 2787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-sbc 2788
This theorem is referenced by: (None)
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