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Theorem reu8 2708
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 2507 . 2 (∃!x A φ∃!y A ψ)
3 reu6 2701 . 2 (∃!y A ψx A y A (ψy = x))
4 dfbi2 368 . . . . 5 ((ψy = x) ↔ ((ψy = x) (y = xψ)))
54ralbii 2302 . . . 4 (y A (ψy = x) ↔ y A ((ψy = x) (y = xψ)))
6 ancom 253 . . . . . 6 ((φ y A (ψx = y)) ↔ (y A (ψx = y) φ))
7 equcom 1569 . . . . . . . . . 10 (x = yy = x)
87imbi2i 215 . . . . . . . . 9 ((ψx = y) ↔ (ψy = x))
98ralbii 2302 . . . . . . . 8 (y A (ψx = y) ↔ y A (ψy = x))
109a1i 9 . . . . . . 7 (x A → (y A (ψx = y) ↔ y A (ψy = x)))
11 biimt 230 . . . . . . . 8 (x A → (φ ↔ (x Aφ)))
12 df-ral 2283 . . . . . . . . 9 (y A (y = xψ) ↔ y(y A → (y = xψ)))
13 bi2.04 237 . . . . . . . . . 10 ((y A → (y = xψ)) ↔ (y = x → (y Aψ)))
1413albii 1335 . . . . . . . . 9 (y(y A → (y = xψ)) ↔ y(y = x → (y Aψ)))
15 vex 2532 . . . . . . . . . 10 x V
16 eleq1 2076 . . . . . . . . . . . . 13 (x = y → (x Ay A))
1716, 1imbi12d 223 . . . . . . . . . . . 12 (x = y → ((x Aφ) ↔ (y Aψ)))
1817bicomd 129 . . . . . . . . . . 11 (x = y → ((y Aψ) ↔ (x Aφ)))
1918equcoms 1570 . . . . . . . . . 10 (y = x → ((y Aψ) ↔ (x Aφ)))
2015, 19ceqsalv 2555 . . . . . . . . 9 (y(y = x → (y Aψ)) ↔ (x Aφ))
2112, 14, 203bitrri 196 . . . . . . . 8 ((x Aφ) ↔ y A (y = xψ))
2211, 21syl6bb 185 . . . . . . 7 (x A → (φy A (y = xψ)))
2310, 22anbi12d 442 . . . . . 6 (x A → ((y A (ψx = y) φ) ↔ (y A (ψy = x) y A (y = xψ))))
246, 23syl5bb 181 . . . . 5 (x A → ((φ y A (ψx = y)) ↔ (y A (ψy = x) y A (y = xψ))))
25 r19.26 2413 . . . . 5 (y A ((ψy = x) (y = xψ)) ↔ (y A (ψy = x) y A (y = xψ)))
2624, 25syl6rbbr 188 . . . 4 (x A → (y A ((ψy = x) (y = xψ)) ↔ (φ y A (ψx = y))))
275, 26syl5bb 181 . . 3 (x A → (y A (ψy = x) ↔ (φ y A (ψx = y))))
2827rexbiia 2311 . 2 (x A y A (ψy = x) ↔ x A (φ y A (ψx = y)))
292, 3, 283bitri 195 1 (∃!x A φx A (φ y A (ψx = y)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1224   wcel 1369  wral 2278  wrex 2279  ∃!wreu 2280
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1622  df-eu 1879  df-clab 2003  df-cleq 2009  df-clel 2012  df-ral 2283  df-rex 2284  df-reu 2285  df-v 2531
This theorem is referenced by: (None)
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