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Theorem reu8 3032
 Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
Hypothesis
Ref Expression
rmo4.1 (x = y → (φψ))
Assertion
Ref Expression
reu8 (∃!x A φx A (φ y A (ψx = y)))
Distinct variable groups:   x,y,A   φ,y   ψ,x
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem reu8
StepHypRef Expression
1 rmo4.1 . . 3 (x = y → (φψ))
21cbvreuv 2837 . 2 (∃!x A φ∃!y A ψ)
3 reu6 3025 . 2 (∃!y A ψx A y A (ψy = x))
4 dfbi2 609 . . . . 5 ((ψy = x) ↔ ((ψy = x) (y = xψ)))
54ralbii 2638 . . . 4 (y A (ψy = x) ↔ y A ((ψy = x) (y = xψ)))
6 ancom 437 . . . . . 6 ((φ y A (ψx = y)) ↔ (y A (ψx = y) φ))
7 equcom 1680 . . . . . . . . . 10 (x = yy = x)
87imbi2i 303 . . . . . . . . 9 ((ψx = y) ↔ (ψy = x))
98ralbii 2638 . . . . . . . 8 (y A (ψx = y) ↔ y A (ψy = x))
109a1i 10 . . . . . . 7 (x A → (y A (ψx = y) ↔ y A (ψy = x)))
11 biimt 325 . . . . . . . 8 (x A → (φ ↔ (x Aφ)))
12 df-ral 2619 . . . . . . . . 9 (y A (y = xψ) ↔ y(y A → (y = xψ)))
13 bi2.04 350 . . . . . . . . . 10 ((y A → (y = xψ)) ↔ (y = x → (y Aψ)))
1413albii 1566 . . . . . . . . 9 (y(y A → (y = xψ)) ↔ y(y = x → (y Aψ)))
15 vex 2862 . . . . . . . . . 10 x V
16 eleq1 2413 . . . . . . . . . . . . 13 (x = y → (x Ay A))
1716, 1imbi12d 311 . . . . . . . . . . . 12 (x = y → ((x Aφ) ↔ (y Aψ)))
1817bicomd 192 . . . . . . . . . . 11 (x = y → ((y Aψ) ↔ (x Aφ)))
1918equcoms 1681 . . . . . . . . . 10 (y = x → ((y Aψ) ↔ (x Aφ)))
2015, 19ceqsalv 2885 . . . . . . . . 9 (y(y = x → (y Aψ)) ↔ (x Aφ))
2112, 14, 203bitrri 263 . . . . . . . 8 ((x Aφ) ↔ y A (y = xψ))
2211, 21syl6bb 252 . . . . . . 7 (x A → (φy A (y = xψ)))
2310, 22anbi12d 691 . . . . . 6 (x A → ((y A (ψx = y) φ) ↔ (y A (ψy = x) y A (y = xψ))))
246, 23syl5bb 248 . . . . 5 (x A → ((φ y A (ψx = y)) ↔ (y A (ψy = x) y A (y = xψ))))
25 r19.26 2746 . . . . 5 (y A ((ψy = x) (y = xψ)) ↔ (y A (ψy = x) y A (y = xψ)))
2624, 25syl6rbbr 255 . . . 4 (x A → (y A ((ψy = x) (y = xψ)) ↔ (φ y A (ψx = y))))
275, 26syl5bb 248 . . 3 (x A → (y A (ψy = x) ↔ (φ y A (ψx = y))))
2827rexbiia 2647 . 2 (x A y A (ψy = x) ↔ x A (φ y A (ψx = y)))
292, 3, 283bitri 262 1 (∃!x A φx A (φ y A (ψx = y)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  ∃!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-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2619  df-rex 2620  df-reu 2621  df-v 2861 This theorem is referenced by: (None)
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