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Theorem sbcralg 2836
 Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcralg (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcralg
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2767 . 2 (𝑧 = 𝐴 → ([𝑧 / 𝑥]∀𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑))
2 dfsbcq2 2767 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32ralbidv 2326 . 2 (𝑧 = 𝐴 → (∀𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
4 nfcv 2178 . . . 4 𝑥𝐵
5 nfs1v 1815 . . . 4 𝑥[𝑧 / 𝑥]𝜑
64, 5nfralxy 2360 . . 3 𝑥𝑦𝐵 [𝑧 / 𝑥]𝜑
7 sbequ12 1654 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
87ralbidv 2326 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑))
96, 8sbie 1674 . 2 ([𝑧 / 𝑥]∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑)
101, 3, 9vtoclbg 2614 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  [wsb 1645  ∀wral 2306  [wsbc 2764 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765 This theorem is referenced by:  r19.12sn  3436
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