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Theorem cbvreucsf 3200
 Description: A more general version of cbvreuv 2837 that has no distinct variable rextrictions. 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 (∃!x A φ∃!y B ψ)

Proof of Theorem cbvreucsf
Dummy variables v z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . 4 z(x A φ)
2 nfcsb1v 3168 . . . . . 6 x[z / x]A
32nfcri 2483 . . . . 5 x z [z / x]A
4 nfs1v 2106 . . . . 5 x[z / x]φ
53, 4nfan 1824 . . . 4 x(z [z / x]A [z / x]φ)
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 sbequ12 1919 . . . . 5 (x = z → (φ ↔ [z / x]φ))
108, 9anbi12d 691 . . . 4 (x = z → ((x A φ) ↔ (z [z / x]A [z / x]φ)))
111, 5, 10cbveu 2224 . . 3 (∃!x(x A φ) ↔ ∃!z(z [z / x]A [z / x]φ))
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
1716nfsb 2109 . . . . 5 y[z / x]φ
1815, 17nfan 1824 . . . 4 y(z [z / x]A [z / x]φ)
19 nfv 1619 . . . 4 z(y B ψ)
20 id 19 . . . . . 6 (z = yz = y)
21 csbeq1 3139 . . . . . . 7 (z = y[z / x]A = [y / x]A)
22 sbsbc 3050 . . . . . . . . 9 ([y / x]v A ↔ [̣y / xv A)
2322abbii 2465 . . . . . . . 8 {v [y / x]v A} = {v y / xv A}
24 cbvralcsf.2 . . . . . . . . . . . 12 xB
2524nfcri 2483 . . . . . . . . . . 11 x v B
26 cbvralcsf.5 . . . . . . . . . . . 12 (x = yA = B)
2726eleq2d 2420 . . . . . . . . . . 11 (x = y → (v Av B))
2825, 27sbie 2038 . . . . . . . . . 10 ([y / x]v Av B)
2928bicomi 193 . . . . . . . . 9 (v B ↔ [y / x]v A)
3029abbi2i 2464 . . . . . . . 8 B = {v [y / x]v A}
31 df-csb 3137 . . . . . . . 8 [y / x]A = {v y / xv A}
3223, 30, 313eqtr4ri 2384 . . . . . . 7 [y / x]A = B
3321, 32syl6eq 2401 . . . . . 6 (z = y[z / x]A = B)
3420, 33eleq12d 2421 . . . . 5 (z = y → (z [z / x]Ay B))
35 sbequ 2060 . . . . . 6 (z = y → ([z / x]φ ↔ [y / x]φ))
36 cbvralcsf.4 . . . . . . 7 xψ
37 cbvralcsf.6 . . . . . . 7 (x = y → (φψ))
3836, 37sbie 2038 . . . . . 6 ([y / x]φψ)
3935, 38syl6bb 252 . . . . 5 (z = y → ([z / x]φψ))
4034, 39anbi12d 691 . . . 4 (z = y → ((z [z / x]A [z / x]φ) ↔ (y B ψ)))
4118, 19, 40cbveu 2224 . . 3 (∃!z(z [z / x]A [z / x]φ) ↔ ∃!y(y B ψ))
4211, 41bitri 240 . 2 (∃!x(x A φ) ↔ ∃!y(y B ψ))
43 df-reu 2621 . 2 (∃!x A φ∃!x(x A φ))
44 df-reu 2621 . 2 (∃!y B ψ∃!y(y B ψ))
4542, 43, 443bitr4i 268 1 (∃!x A φ∃!y B ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃!weu 2204  {cab 2339  Ⅎwnfc 2476  ∃!wreu 2616  [̣wsbc 3046  [csb 3136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-reu 2621  df-sbc 3047  df-csb 3137 This theorem is referenced by: (None)
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