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Theorem cbvreu 2833
 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 1619 . . . 4 z(x A φ)
21sb8eu 2222 . . 3 (∃!x(x A φ) ↔ ∃!z[z / x](x A φ))
3 sban 2069 . . . 4 ([z / x](x A φ) ↔ ([z / x]x A [z / x]φ))
43eubii 2213 . . 3 (∃!z[z / x](x A φ) ↔ ∃!z([z / x]x A [z / x]φ))
5 clelsb3 2455 . . . . . 6 ([z / x]x Az A)
65anbi1i 676 . . . . 5 (([z / x]x A [z / x]φ) ↔ (z A [z / x]φ))
76eubii 2213 . . . 4 (∃!z([z / x]x A [z / x]φ) ↔ ∃!z(z A [z / x]φ))
8 nfv 1619 . . . . . 6 y z A
9 cbvral.1 . . . . . . 7 yφ
109nfsb 2109 . . . . . 6 y[z / x]φ
118, 10nfan 1824 . . . . 5 y(z A [z / x]φ)
12 nfv 1619 . . . . 5 z(y A ψ)
13 eleq1 2413 . . . . . 6 (z = y → (z Ay A))
14 sbequ 2060 . . . . . . 7 (z = y → ([z / x]φ ↔ [y / x]φ))
15 cbvral.2 . . . . . . . 8 xψ
16 cbvral.3 . . . . . . . 8 (x = y → (φψ))
1715, 16sbie 2038 . . . . . . 7 ([y / x]φψ)
1814, 17syl6bb 252 . . . . . 6 (z = y → ([z / x]φψ))
1913, 18anbi12d 691 . . . . 5 (z = y → ((z A [z / x]φ) ↔ (y A ψ)))
2011, 12, 19cbveu 2224 . . . 4 (∃!z(z A [z / x]φ) ↔ ∃!y(y A ψ))
217, 20bitri 240 . . 3 (∃!z([z / x]x A [z / x]φ) ↔ ∃!y(y A ψ))
222, 4, 213bitri 262 . 2 (∃!x(x A φ) ↔ ∃!y(y A ψ))
23 df-reu 2621 . 2 (∃!x A φ∃!x(x A φ))
24 df-reu 2621 . 2 (∃!y A ψ∃!y(y A ψ))
2522, 23, 243bitr4i 268 1 (∃!x A φ∃!y A 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  ∃!wreu 2616 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-eu 2208  df-cleq 2346  df-clel 2349  df-reu 2621 This theorem is referenced by:  cbvrmo  2834  cbvreuv  2837
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