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Theorem cbvralcsf 3198
 Description: A more general version of cbvralf 2829 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 (x A φy B ψ)

Proof of Theorem cbvralcsf
Dummy variables v z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . 4 z(x Aφ)
2 nfcsb1v 3168 . . . . . 6 x[z / x]A
32nfcri 2483 . . . . 5 x z [z / x]A
4 nfsbc1v 3065 . . . . 5 xz / xφ
53, 4nfim 1813 . . . 4 x(z [z / x]A → [̣z / xφ)
6 id 19 . . . . . 6 (x = zx = z)
7 csbeq1a 3144 . . . . . 6 (x = zA = [z / x]A)
86, 7eleq12d 2421 . . . . 5 (x = z → (x Az [z / x]A))
9 sbceq1a 3056 . . . . 5 (x = z → (φ ↔ [̣z / xφ))
108, 9imbi12d 311 . . . 4 (x = z → ((x Aφ) ↔ (z [z / x]A → [̣z / xφ)))
111, 5, 10cbval 1984 . . 3 (x(x Aφ) ↔ z(z [z / x]A → [̣z / xφ))
12 nfcv 2489 . . . . . . 7 yz
13 cbvralcsf.1 . . . . . . 7 yA
1412, 13nfcsb 3170 . . . . . 6 y[z / x]A
1514nfcri 2483 . . . . 5 y z [z / x]A
16 cbvralcsf.3 . . . . . 6 yφ
1712, 16nfsbc 3067 . . . . 5 yz / xφ
1815, 17nfim 1813 . . . 4 y(z [z / x]A → [̣z / xφ)
19 nfv 1619 . . . 4 z(y Bψ)
20 id 19 . . . . . 6 (z = yz = y)
21 csbeq1 3139 . . . . . . 7 (z = y[z / x]A = [y / x]A)
22 df-csb 3137 . . . . . . . 8 [y / x]A = {v y / xv A}
23 cbvralcsf.2 . . . . . . . . . . . 12 xB
2423nfcri 2483 . . . . . . . . . . 11 x v B
25 cbvralcsf.5 . . . . . . . . . . . 12 (x = yA = B)
2625eleq2d 2420 . . . . . . . . . . 11 (x = y → (v Av B))
2724, 26sbie 2038 . . . . . . . . . 10 ([y / x]v Av B)
28 sbsbc 3050 . . . . . . . . . 10 ([y / x]v A ↔ [̣y / xv A)
2927, 28bitr3i 242 . . . . . . . . 9 (v B ↔ [̣y / xv A)
3029abbi2i 2464 . . . . . . . 8 B = {v y / xv A}
3122, 30eqtr4i 2376 . . . . . . 7 [y / x]A = B
3221, 31syl6eq 2401 . . . . . 6 (z = y[z / x]A = B)
3320, 32eleq12d 2421 . . . . 5 (z = y → (z [z / x]Ay B))
34 dfsbcq 3048 . . . . . 6 (z = y → ([̣z / xφ ↔ [̣y / xφ))
35 sbsbc 3050 . . . . . . 7 ([y / x]φ ↔ [̣y / xφ)
36 cbvralcsf.4 . . . . . . . 8 xψ
37 cbvralcsf.6 . . . . . . . 8 (x = y → (φψ))
3836, 37sbie 2038 . . . . . . 7 ([y / x]φψ)
3935, 38bitr3i 242 . . . . . 6 ([̣y / xφψ)
4034, 39syl6bb 252 . . . . 5 (z = y → ([̣z / xφψ))
4133, 40imbi12d 311 . . . 4 (z = y → ((z [z / x]A → [̣z / xφ) ↔ (y Bψ)))
4218, 19, 41cbval 1984 . . 3 (z(z [z / x]A → [̣z / xφ) ↔ y(y Bψ))
4311, 42bitri 240 . 2 (x(x Aφ) ↔ y(y Bψ))
44 df-ral 2619 . 2 (x A φx(x Aφ))
45 df-ral 2619 . 2 (y B ψy(y Bψ))
4643, 44, 453bitr4i 268 1 (x A φy B ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476  ∀wral 2614  [̣wsbc 3046  [csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-sbc 3047  df-csb 3137 This theorem is referenced by:  cbvrexcsf  3199  cbvralv2  3202
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