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Theorem cbvrexcsf 2976
Description: A more general version of cbvrexf 2578 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 1462 . . . 4 𝑧(𝑥𝐴𝜑)
2 nfcsb1v 2949 . . . . . 6 𝑥𝑧 / 𝑥𝐴
32nfcri 2217 . . . . 5 𝑥 𝑧𝑧 / 𝑥𝐴
4 nfsbc1v 2844 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfan 1498 . . . 4 𝑥(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑)
6 id 19 . . . . . 6 (𝑥 = 𝑧𝑥 = 𝑧)
7 csbeq1a 2927 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
86, 7eleq12d 2153 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
9 sbceq1a 2835 . . . . 5 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
108, 9anbi12d 457 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑)))
111, 5, 10cbvex 1681 . . 3 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑧(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑))
12 nfcv 2223 . . . . . . 7 𝑦𝑧
13 cbvralcsf.1 . . . . . . 7 𝑦𝐴
1412, 13nfcsb 2951 . . . . . 6 𝑦𝑧 / 𝑥𝐴
1514nfcri 2217 . . . . 5 𝑦 𝑧𝑧 / 𝑥𝐴
16 cbvralcsf.3 . . . . . 6 𝑦𝜑
1712, 16nfsbc 2846 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1815, 17nfan 1498 . . . 4 𝑦(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑)
19 nfv 1462 . . . 4 𝑧(𝑦𝐵𝜓)
20 id 19 . . . . . 6 (𝑧 = 𝑦𝑧 = 𝑦)
21 csbeq1 2922 . . . . . . 7 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝑦 / 𝑥𝐴)
22 df-csb 2920 . . . . . . . 8 𝑦 / 𝑥𝐴 = {𝑣[𝑦 / 𝑥]𝑣𝐴}
23 cbvralcsf.2 . . . . . . . . . . . 12 𝑥𝐵
2423nfcri 2217 . . . . . . . . . . 11 𝑥 𝑣𝐵
25 cbvralcsf.5 . . . . . . . . . . . 12 (𝑥 = 𝑦𝐴 = 𝐵)
2625eleq2d 2152 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑣𝐴𝑣𝐵))
2724, 26sbie 1716 . . . . . . . . . 10 ([𝑦 / 𝑥]𝑣𝐴𝑣𝐵)
28 sbsbc 2830 . . . . . . . . . 10 ([𝑦 / 𝑥]𝑣𝐴[𝑦 / 𝑥]𝑣𝐴)
2927, 28bitr3i 184 . . . . . . . . 9 (𝑣𝐵[𝑦 / 𝑥]𝑣𝐴)
3029abbi2i 2197 . . . . . . . 8 𝐵 = {𝑣[𝑦 / 𝑥]𝑣𝐴}
3122, 30eqtr4i 2106 . . . . . . 7 𝑦 / 𝑥𝐴 = 𝐵
3221, 31syl6eq 2131 . . . . . 6 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝐵)
3320, 32eleq12d 2153 . . . . 5 (𝑧 = 𝑦 → (𝑧𝑧 / 𝑥𝐴𝑦𝐵))
34 dfsbcq 2828 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
35 sbsbc 2830 . . . . . . 7 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
36 cbvralcsf.4 . . . . . . . 8 𝑥𝜓
37 cbvralcsf.6 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
3836, 37sbie 1716 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜓)
3935, 38bitr3i 184 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
4034, 39syl6bb 194 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
4133, 40anbi12d 457 . . . 4 (𝑧 = 𝑦 → ((𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑) ↔ (𝑦𝐵𝜓)))
4218, 19, 41cbvex 1681 . . 3 (∃𝑧(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑) ↔ ∃𝑦(𝑦𝐵𝜓))
4311, 42bitri 182 . 2 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑦(𝑦𝐵𝜓))
44 df-rex 2359 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
45 df-rex 2359 . 2 (∃𝑦𝐵 𝜓 ↔ ∃𝑦(𝑦𝐵𝜓))
4643, 44, 453bitr4i 210 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wnf 1390  wex 1422  wcel 1434  [wsb 1687  {cab 2069  wnfc 2210  wrex 2354  [wsbc 2826  csb 2919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-sbc 2827  df-csb 2920
This theorem is referenced by:  cbvrexv2  2980
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