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Theorem cbvralcsf 3558
Description: A more general version of cbvralf 3160 that doesn't require 𝐴 and 𝐵 to be distinct from 𝑥 or 𝑦. 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
cbvralcsf (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐵 𝜓)

Proof of Theorem cbvralcsf
Dummy variables 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1841 . . . 4 𝑧(𝑥𝐴𝜑)
2 nfcsb1v 3542 . . . . . 6 𝑥𝑧 / 𝑥𝐴
32nfcri 2756 . . . . 5 𝑥 𝑧𝑧 / 𝑥𝐴
4 nfsbc1v 3449 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfim 1823 . . . 4 𝑥(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑)
6 id 22 . . . . . 6 (𝑥 = 𝑧𝑥 = 𝑧)
7 csbeq1a 3535 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
86, 7eleq12d 2693 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
9 sbceq1a 3440 . . . . 5 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
108, 9imbi12d 334 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑)))
111, 5, 10cbval 2269 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑧(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑))
12 nfcv 2762 . . . . . . 7 𝑦𝑧
13 cbvralcsf.1 . . . . . . 7 𝑦𝐴
1412, 13nfcsb 3544 . . . . . 6 𝑦𝑧 / 𝑥𝐴
1514nfcri 2756 . . . . 5 𝑦 𝑧𝑧 / 𝑥𝐴
16 cbvralcsf.3 . . . . . 6 𝑦𝜑
1712, 16nfsbc 3451 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1815, 17nfim 1823 . . . 4 𝑦(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑)
19 nfv 1841 . . . 4 𝑧(𝑦𝐵𝜓)
20 id 22 . . . . . 6 (𝑧 = 𝑦𝑧 = 𝑦)
21 csbeq1 3529 . . . . . . 7 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝑦 / 𝑥𝐴)
22 df-csb 3527 . . . . . . . 8 𝑦 / 𝑥𝐴 = {𝑣[𝑦 / 𝑥]𝑣𝐴}
23 cbvralcsf.2 . . . . . . . . . . . 12 𝑥𝐵
2423nfcri 2756 . . . . . . . . . . 11 𝑥 𝑣𝐵
25 cbvralcsf.5 . . . . . . . . . . . 12 (𝑥 = 𝑦𝐴 = 𝐵)
2625eleq2d 2685 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑣𝐴𝑣𝐵))
2724, 26sbie 2406 . . . . . . . . . 10 ([𝑦 / 𝑥]𝑣𝐴𝑣𝐵)
28 sbsbc 3433 . . . . . . . . . 10 ([𝑦 / 𝑥]𝑣𝐴[𝑦 / 𝑥]𝑣𝐴)
2927, 28bitr3i 266 . . . . . . . . 9 (𝑣𝐵[𝑦 / 𝑥]𝑣𝐴)
3029abbi2i 2736 . . . . . . . 8 𝐵 = {𝑣[𝑦 / 𝑥]𝑣𝐴}
3122, 30eqtr4i 2645 . . . . . . 7 𝑦 / 𝑥𝐴 = 𝐵
3221, 31syl6eq 2670 . . . . . 6 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝐵)
3320, 32eleq12d 2693 . . . . 5 (𝑧 = 𝑦 → (𝑧𝑧 / 𝑥𝐴𝑦𝐵))
34 dfsbcq 3431 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑[𝑦 / 𝑥]𝜑))
35 sbsbc 3433 . . . . . . 7 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
36 cbvralcsf.4 . . . . . . . 8 𝑥𝜓
37 cbvralcsf.6 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
3836, 37sbie 2406 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜓)
3935, 38bitr3i 266 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
4034, 39syl6bb 276 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
4133, 40imbi12d 334 . . . 4 (𝑧 = 𝑦 → ((𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑) ↔ (𝑦𝐵𝜓)))
4218, 19, 41cbval 2269 . . 3 (∀𝑧(𝑧𝑧 / 𝑥𝐴[𝑧 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐵𝜓))
4311, 42bitri 264 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑦(𝑦𝐵𝜓))
44 df-ral 2914 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
45 df-ral 2914 . 2 (∀𝑦𝐵 𝜓 ↔ ∀𝑦(𝑦𝐵𝜓))
4643, 44, 453bitr4i 292 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1479   = wceq 1481  wnf 1706  [wsb 1878  wcel 1988  {cab 2606  wnfc 2749  wral 2909  [wsbc 3429  csb 3526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-sbc 3430  df-csb 3527
This theorem is referenced by:  cbvrexcsf  3559  cbvralv2  3562
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