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Theorem sbcralt 2955
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 2901 . 2 ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑)
2 simpl 108 . . 3 ((𝐴𝑉𝑦𝐴) → 𝐴𝑉)
3 sbsbc 2884 . . . . 5 ([𝑧 / 𝑥]∀𝑦𝐵 𝜑[𝑧 / 𝑥]𝑦𝐵 𝜑)
4 nfcv 2256 . . . . . . 7 𝑥𝐵
5 nfs1v 1890 . . . . . . 7 𝑥[𝑧 / 𝑥]𝜑
64, 5nfralxy 2446 . . . . . 6 𝑥𝑦𝐵 [𝑧 / 𝑥]𝜑
7 sbequ12 1727 . . . . . . 7 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
87ralbidv 2412 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑))
96, 8sbie 1747 . . . . 5 ([𝑧 / 𝑥]∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑)
103, 9bitr3i 185 . . . 4 ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑)
11 nfnfc1 2259 . . . . . . 7 𝑦𝑦𝐴
12 nfcvd 2257 . . . . . . . 8 (𝑦𝐴𝑦𝑧)
13 id 19 . . . . . . . 8 (𝑦𝐴𝑦𝐴)
1412, 13nfeqd 2271 . . . . . . 7 (𝑦𝐴 → Ⅎ𝑦 𝑧 = 𝐴)
1511, 14nfan1 1526 . . . . . 6 𝑦(𝑦𝐴𝑧 = 𝐴)
16 dfsbcq2 2883 . . . . . . 7 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
1716adantl 273 . . . . . 6 ((𝑦𝐴𝑧 = 𝐴) → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
1815, 17ralbid 2410 . . . . 5 ((𝑦𝐴𝑧 = 𝐴) → (∀𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
1918adantll 465 . . . 4 (((𝐴𝑉𝑦𝐴) ∧ 𝑧 = 𝐴) → (∀𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
2010, 19syl5bb 191 . . 3 (((𝐴𝑉𝑦𝐴) ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
212, 20sbcied 2915 . 2 ((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
221, 21syl5bbr 193 1 ((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  [wsb 1718  wnfc 2243  wral 2391  [wsbc 2880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-sbc 2881
This theorem is referenced by: (None)
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