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Theorem sbcrext 2832
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 2769 . . 3 ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V)
21a1i 9 . 2 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V))
3 nfnfc1 2181 . . 3 𝑦𝑦𝐴
4 id 19 . . . 4 (𝑦𝐴𝑦𝐴)
5 nfcvd 2179 . . . 4 (𝑦𝐴𝑦V)
64, 5nfeld 2193 . . 3 (𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
7 sbcex 2769 . . . 4 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
872a1i 24 . . 3 (𝑦𝐴 → (𝑦𝐵 → ([𝐴 / 𝑥]𝜑𝐴 ∈ V)))
93, 6, 8rexlimd2 2428 . 2 (𝑦𝐴 → (∃𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V))
10 sbcco 2782 . . . 4 ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑[𝐴 / 𝑥]𝑦𝐵 𝜑)
11 simpl 102 . . . . 5 ((𝐴 ∈ V ∧ 𝑦𝐴) → 𝐴 ∈ V)
12 sbsbc 2765 . . . . . . 7 ([𝑧 / 𝑥]∃𝑦𝐵 𝜑[𝑧 / 𝑥]𝑦𝐵 𝜑)
13 nfcv 2178 . . . . . . . . 9 𝑥𝐵
14 nfs1v 1815 . . . . . . . . 9 𝑥[𝑧 / 𝑥]𝜑
1513, 14nfrexxy 2358 . . . . . . . 8 𝑥𝑦𝐵 [𝑧 / 𝑥]𝜑
16 sbequ12 1654 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
1716rexbidv 2324 . . . . . . . 8 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑))
1815, 17sbie 1674 . . . . . . 7 ([𝑧 / 𝑥]∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
1912, 18bitr3i 175 . . . . . 6 ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝑧 / 𝑥]𝜑)
20 nfcvd 2179 . . . . . . . . . 10 (𝑦𝐴𝑦𝑧)
2120, 4nfeqd 2192 . . . . . . . . 9 (𝑦𝐴 → Ⅎ𝑦 𝑧 = 𝐴)
223, 21nfan1 1456 . . . . . . . 8 𝑦(𝑦𝐴𝑧 = 𝐴)
23 dfsbcq2 2764 . . . . . . . . 9 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2423adantl 262 . . . . . . . 8 ((𝑦𝐴𝑧 = 𝐴) → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2522, 24rexbid 2322 . . . . . . 7 ((𝑦𝐴𝑧 = 𝐴) → (∃𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2625adantll 445 . . . . . 6 (((𝐴 ∈ V ∧ 𝑦𝐴) ∧ 𝑧 = 𝐴) → (∃𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2719, 26syl5bb 181 . . . . 5 (((𝐴 ∈ V ∧ 𝑦𝐴) ∧ 𝑧 = 𝐴) → ([𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2811, 27sbcied 2796 . . . 4 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑧][𝑧 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2910, 28syl5bbr 183 . . 3 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
3029expcom 109 . 2 (𝑦𝐴 → (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)))
312, 9, 30pm5.21ndd 621 1 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  [wsb 1645  wnfc 2165  wrex 2304  Vcvv 2554  [wsbc 2761
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-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762
This theorem is referenced by:  sbcrex  2834
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