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Theorem sbcralt 3054
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 2999 . 2 ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑)
2 simpl 109 . . 3 ((𝐴𝑉𝑦𝐴) → 𝐴𝑉)
3 sbsbc 2981 . . . . 5 ([𝑧 / 𝑥]∀𝑦𝐵 𝜑[𝑧 / 𝑥]𝑦𝐵 𝜑)
4 nfcv 2332 . . . . . . 7 𝑥𝐵
5 nfs1v 1951 . . . . . . 7 𝑥[𝑧 / 𝑥]𝜑
64, 5nfralxy 2528 . . . . . 6 𝑥𝑦𝐵 [𝑧 / 𝑥]𝜑
7 sbequ12 1782 . . . . . . 7 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
87ralbidv 2490 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑))
96, 8sbie 1802 . . . . 5 ([𝑧 / 𝑥]∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑)
103, 9bitr3i 186 . . . 4 ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝑧 / 𝑥]𝜑)
11 nfnfc1 2335 . . . . . . 7 𝑦𝑦𝐴
12 nfcvd 2333 . . . . . . . 8 (𝑦𝐴𝑦𝑧)
13 id 19 . . . . . . . 8 (𝑦𝐴𝑦𝐴)
1412, 13nfeqd 2347 . . . . . . 7 (𝑦𝐴 → Ⅎ𝑦 𝑧 = 𝐴)
1511, 14nfan1 1575 . . . . . 6 𝑦(𝑦𝐴𝑧 = 𝐴)
16 dfsbcq2 2980 . . . . . . 7 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
1716adantl 277 . . . . . 6 ((𝑦𝐴𝑧 = 𝐴) → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
1815, 17ralbid 2488 . . . . 5 ((𝑦𝐴𝑧 = 𝐴) → (∀𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
1918adantll 476 . . . 4 (((𝐴𝑉𝑦𝐴) ∧ 𝑧 = 𝐴) → (∀𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
2010, 19bitrid 192 . . 3 (((𝐴𝑉𝑦𝐴) ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
212, 20sbcied 3014 . 2 ((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
221, 21bitr3id 194 1 ((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  [wsb 1773  wcel 2160  wnfc 2319  wral 2468  [wsbc 2977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-sbc 2978
This theorem is referenced by: (None)
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