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Theorem cbvrexcsf 2906
Description: A more general version of cbvrexf 2525 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
cbvralcsf.1 𝑦𝐴
cbvralcsf.2 𝑥𝐵
cbvralcsf.3 𝑦𝜑
cbvralcsf.4 𝑥𝜓
cbvralcsf.5 (𝑥 = 𝑦𝐴 = 𝐵)
cbvralcsf.6 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrexcsf (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓)

Proof of Theorem cbvrexcsf
Dummy variables 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1421 . . . 4 𝑧(𝑥𝐴𝜑)
2 nfcsb1v 2879 . . . . . 6 𝑥𝑧 / 𝑥𝐴
32nfcri 2172 . . . . 5 𝑥 𝑧𝑧 / 𝑥𝐴
4 nfsbc1v 2779 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfan 1457 . . . 4 𝑥(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑)
6 id 19 . . . . . 6 (𝑥 = 𝑧𝑥 = 𝑧)
7 csbeq1a 2857 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
86, 7eleq12d 2108 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
9 sbceq1a 2770 . . . . 5 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
108, 9anbi12d 442 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑)))
111, 5, 10cbvex 1639 . . 3 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑))
12 nfcv 2178 . . . . . . 7 𝑦𝑧
13 cbvralcsf.1 . . . . . . 7 𝑦𝐴
1412, 13nfcsb 2881 . . . . . 6 𝑦𝑧 / 𝑥𝐴
1514nfcri 2172 . . . . 5 𝑦 𝑧𝑧 / 𝑥𝐴
16 cbvralcsf.3 . . . . . 6 𝑦𝜑
1712, 16nfsbc 2781 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1815, 17nfan 1457 . . . 4 𝑦(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑)
19 nfv 1421 . . . 4 𝑧(𝑦𝐵𝜓)
20 id 19 . . . . . 6 (𝑧 = 𝑦𝑧 = 𝑦)
21 csbeq1 2852 . . . . . . 7 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝑦 / 𝑥𝐴)
22 df-csb 2850 . . . . . . . 8 𝑦 / 𝑥𝐴 = {𝑣[𝑦 / 𝑥]𝑣𝐴}
23 cbvralcsf.2 . . . . . . . . . . . 12 𝑥𝐵
2423nfcri 2172 . . . . . . . . . . 11 𝑥 𝑣𝐵
25 cbvralcsf.5 . . . . . . . . . . . 12 (𝑥 = 𝑦𝐴 = 𝐵)
2625eleq2d 2107 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑣𝐴𝑣𝐵))
2724, 26sbie 1674 . . . . . . . . . 10 ([𝑦 / 𝑥]𝑣𝐴𝑣𝐵)
28 sbsbc 2765 . . . . . . . . . 10 ([𝑦 / 𝑥]𝑣𝐴[𝑦 / 𝑥]𝑣𝐴)
2927, 28bitr3i 175 . . . . . . . . 9 (𝑣𝐵[𝑦 / 𝑥]𝑣𝐴)
3029abbi2i 2152 . . . . . . . 8 𝐵 = {𝑣[𝑦 / 𝑥]𝑣𝐴}
3122, 30eqtr4i 2063 . . . . . . 7 𝑦 / 𝑥𝐴 = 𝐵
3221, 31syl6eq 2088 . . . . . 6 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝐵)
3320, 32eleq12d 2108 . . . . 5 (𝑧 = 𝑦 → (𝑧𝑧 / 𝑥𝐴𝑦𝐵))
34 dfsbcq 2763 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
35 sbsbc 2765 . . . . . . 7 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
36 cbvralcsf.4 . . . . . . . 8 𝑥𝜓
37 cbvralcsf.6 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
3836, 37sbie 1674 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜓)
3935, 38bitr3i 175 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
4034, 39syl6bb 185 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
4133, 40anbi12d 442 . . . 4 (𝑧 = 𝑦 → ((𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑) ↔ (𝑦𝐵𝜓)))
4218, 19, 41cbvex 1639 . . 3 (∃𝑧(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑) ↔ ∃𝑦(𝑦𝐵𝜓))
4311, 42bitri 173 . 2 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑦(𝑦𝐵𝜓))
44 df-rex 2309 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
45 df-rex 2309 . 2 (∃𝑦𝐵 𝜓 ↔ ∃𝑦(𝑦𝐵𝜓))
4643, 44, 453bitr4i 201 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wnf 1349  wex 1381  wcel 1393  [wsb 1645  {cab 2026  wnfc 2165  wrex 2304  [wsbc 2761  csb 2849
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-rex 2309  df-sbc 2762  df-csb 2850
This theorem is referenced by:  cbvrexv2  2910
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