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Theorem sbcrext 2986
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbcrext (𝑦𝐴 → ([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)

Proof of Theorem sbcrext
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sbcex 2917 . . 3 ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V)
21a1i 9 . 2 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V))
3 nfnfc1 2284 . . 3 𝑦𝑦𝐴
4 id 19 . . . 4 (𝑦𝐴𝑦𝐴)
5 nfcvd 2282 . . . 4 (𝑦𝐴𝑦V)
64, 5nfeld 2297 . . 3 (𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
7 sbcex 2917 . . . 4 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
872a1i 27 . . 3 (𝑦𝐴 → (𝑦𝐵 → ([𝐴 / 𝑥]𝜑𝐴 ∈ V)))
93, 6, 8rexlimd2 2547 . 2 (𝑦𝐴 → (∃𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V))
10 sbcco 2930 . . . 4 ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑)
11 simpl 108 . . . . 5 ((𝐴 ∈ V ∧ 𝑦𝐴) → 𝐴 ∈ V)
12 sbsbc 2913 . . . . . . 7 ([𝑧 / 𝑥]∃𝑦𝐵 𝜑[𝑧 / 𝑥]𝑦𝐵 𝜑)
13 nfcv 2281 . . . . . . . . 9 𝑥𝐵
14 nfs1v 1912 . . . . . . . . 9 𝑥[𝑧 / 𝑥]𝜑
1513, 14nfrexxy 2472 . . . . . . . 8 𝑥𝑦𝐵 [𝑧 / 𝑥]𝜑
16 sbequ12 1744 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
1716rexbidv 2438 . . . . . . . 8 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑))
1815, 17sbie 1764 . . . . . . 7 ([𝑧 / 𝑥]∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
1912, 18bitr3i 185 . . . . . 6 ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
20 nfcvd 2282 . . . . . . . . . 10 (𝑦𝐴𝑦𝑧)
2120, 4nfeqd 2296 . . . . . . . . 9 (𝑦𝐴 → Ⅎ𝑦 𝑧 = 𝐴)
223, 21nfan1 1543 . . . . . . . 8 𝑦(𝑦𝐴𝑧 = 𝐴)
23 dfsbcq2 2912 . . . . . . . . 9 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2423adantl 275 . . . . . . . 8 ((𝑦𝐴𝑧 = 𝐴) → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2522, 24rexbid 2436 . . . . . . 7 ((𝑦𝐴𝑧 = 𝐴) → (∃𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2625adantll 467 . . . . . 6 (((𝐴 ∈ V ∧ 𝑦𝐴) ∧ 𝑧 = 𝐴) → (∃𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2719, 26syl5bb 191 . . . . 5 (((𝐴 ∈ V ∧ 𝑦𝐴) ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2811, 27sbcied 2945 . . . 4 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2910, 28bitr3id 193 . . 3 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
3029expcom 115 . 2 (𝑦𝐴 → (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)))
312, 9, 30pm5.21ndd 694 1 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  [wsb 1735  wnfc 2268  wrex 2417  Vcvv 2686  [wsbc 2909
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910
This theorem is referenced by:  sbcrex  2988
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