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Theorem cbvreucsf 3713
Description: A more general version of cbvreuv 3320 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
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
cbvreucsf (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐵 𝜓)

Proof of Theorem cbvreucsf
Dummy variables 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1993 . . . 4 𝑧(𝑥𝐴𝜑)
2 nfcsb1v 3695 . . . . . 6 𝑥𝑧 / 𝑥𝐴
32nfcri 2905 . . . . 5 𝑥 𝑧𝑧 / 𝑥𝐴
4 nfs1v 2272 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfan 1978 . . . 4 𝑥(𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)
6 id 22 . . . . . 6 (𝑥 = 𝑧𝑥 = 𝑧)
7 csbeq1a 3688 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
86, 7eleq12d 2842 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
9 sbequ12 2265 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
108, 9anbi12d 746 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)))
111, 5, 10cbveu 2652 . . 3 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑧(𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
12 nfcv 2911 . . . . . . 7 𝑦𝑧
13 cbvralcsf.1 . . . . . . 7 𝑦𝐴
1412, 13nfcsb 3697 . . . . . 6 𝑦𝑧 / 𝑥𝐴
1514nfcri 2905 . . . . 5 𝑦 𝑧𝑧 / 𝑥𝐴
16 cbvralcsf.3 . . . . . 6 𝑦𝜑
1716nfsb 2588 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1815, 17nfan 1978 . . . 4 𝑦(𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)
19 nfv 1993 . . . 4 𝑧(𝑦𝐵𝜓)
20 id 22 . . . . . 6 (𝑧 = 𝑦𝑧 = 𝑦)
21 csbeq1 3682 . . . . . . 7 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝑦 / 𝑥𝐴)
22 sbsbc 3588 . . . . . . . . 9 ([𝑦 / 𝑥]𝑣𝐴[𝑦 / 𝑥]𝑣𝐴)
2322abbii 2886 . . . . . . . 8 {𝑣 ∣ [𝑦 / 𝑥]𝑣𝐴} = {𝑣[𝑦 / 𝑥]𝑣𝐴}
24 cbvralcsf.2 . . . . . . . . . . . 12 𝑥𝐵
2524nfcri 2905 . . . . . . . . . . 11 𝑥 𝑣𝐵
26 cbvralcsf.5 . . . . . . . . . . . 12 (𝑥 = 𝑦𝐴 = 𝐵)
2726eleq2d 2834 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑣𝐴𝑣𝐵))
2825, 27sbie 2553 . . . . . . . . . 10 ([𝑦 / 𝑥]𝑣𝐴𝑣𝐵)
2928bicomi 214 . . . . . . . . 9 (𝑣𝐵 ↔ [𝑦 / 𝑥]𝑣𝐴)
3029abbi2i 2885 . . . . . . . 8 𝐵 = {𝑣 ∣ [𝑦 / 𝑥]𝑣𝐴}
31 df-csb 3680 . . . . . . . 8 𝑦 / 𝑥𝐴 = {𝑣[𝑦 / 𝑥]𝑣𝐴}
3223, 30, 313eqtr4ri 2802 . . . . . . 7 𝑦 / 𝑥𝐴 = 𝐵
3321, 32syl6eq 2819 . . . . . 6 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝐵)
3420, 33eleq12d 2842 . . . . 5 (𝑧 = 𝑦 → (𝑧𝑧 / 𝑥𝐴𝑦𝐵))
35 sbequ 2521 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
36 cbvralcsf.4 . . . . . . 7 𝑥𝜓
37 cbvralcsf.6 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
3836, 37sbie 2553 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
3935, 38syl6bb 276 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
4034, 39anbi12d 746 . . . 4 (𝑧 = 𝑦 → ((𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐵𝜓)))
4118, 19, 40cbveu 2652 . . 3 (∃!𝑧(𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦𝐵𝜓))
4211, 41bitri 264 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑦(𝑦𝐵𝜓))
43 df-reu 3066 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
44 df-reu 3066 . 2 (∃!𝑦𝐵 𝜓 ↔ ∃!𝑦(𝑦𝐵𝜓))
4542, 43, 443bitr4i 292 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1629  wnf 1854  [wsb 2047  wcel 2143  ∃!weu 2616  {cab 2755  wnfc 2898  ∃!wreu 3061  [wsbc 3584  csb 3679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-reu 3066  df-sbc 3585  df-csb 3680
This theorem is referenced by: (None)
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