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Theorem cbvreu 2500
Description: Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
cbvral.1 yφ
cbvral.2 xψ
cbvral.3 (x = y → (φψ))
Assertion
Ref Expression
cbvreu (∃!x A φ∃!y A ψ)
Distinct variable groups:   x,A   y,A
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem cbvreu
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfv 1394 . . . 4 z(x A φ)
21sb8eu 1886 . . 3 (∃!x(x A φ) ↔ ∃!z[z / x](x A φ))
3 sban 1802 . . . 4 ([z / x](x A φ) ↔ ([z / x]x A [z / x]φ))
43eubii 1882 . . 3 (∃!z[z / x](x A φ) ↔ ∃!z([z / x]x A [z / x]φ))
5 clelsb3 2115 . . . . . 6 ([z / x]x Az A)
65anbi1i 431 . . . . 5 (([z / x]x A [z / x]φ) ↔ (z A [z / x]φ))
76eubii 1882 . . . 4 (∃!z([z / x]x A [z / x]φ) ↔ ∃!z(z A [z / x]φ))
8 nfv 1394 . . . . . 6 y z A
9 cbvral.1 . . . . . . 7 yφ
109nfsb 1795 . . . . . 6 y[z / x]φ
118, 10nfan 1430 . . . . 5 y(z A [z / x]φ)
12 nfv 1394 . . . . 5 z(y A ψ)
13 eleq1 2073 . . . . . 6 (z = y → (z Ay A))
14 sbequ 1694 . . . . . . 7 (z = y → ([z / x]φ ↔ [y / x]φ))
15 cbvral.2 . . . . . . . 8 xψ
16 cbvral.3 . . . . . . . 8 (x = y → (φψ))
1715, 16sbie 1647 . . . . . . 7 ([y / x]φψ)
1814, 17syl6bb 185 . . . . . 6 (z = y → ([z / x]φψ))
1913, 18anbi12d 442 . . . . 5 (z = y → ((z A [z / x]φ) ↔ (y A ψ)))
2011, 12, 19cbveu 1897 . . . 4 (∃!z(z A [z / x]φ) ↔ ∃!y(y A ψ))
217, 20bitri 173 . . 3 (∃!z([z / x]x A [z / x]φ) ↔ ∃!y(y A ψ))
222, 4, 213bitri 195 . 2 (∃!x(x A φ) ↔ ∃!y(y A ψ))
23 df-reu 2282 . 2 (∃!x A φ∃!x(x A φ))
24 df-reu 2282 . 2 (∃!y A ψ∃!y(y A ψ))
2522, 23, 243bitr4i 201 1 (∃!x A φ∃!y A ψ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wnf 1322   wcel 1366  [wsb 1618  ∃!weu 1873  ∃!wreu 2277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-cleq 2006  df-clel 2009  df-reu 2282
This theorem is referenced by:  cbvrmo  2501  cbvreuv  2504
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