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Theorem reueq 2706
 Description: Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
Assertion
Ref Expression
reueq (B A∃!x A x = B)
Distinct variable groups:   x,A   x,B

Proof of Theorem reueq
StepHypRef Expression
1 risset 2321 . 2 (B Ax A x = B)
2 moeq 2684 . . . 4 ∃*x x = B
3 mormo 2490 . . . 4 (∃*x x = B∃*x A x = B)
42, 3ax-mp 7 . . 3 ∃*x A x = B
5 reu5 2491 . . 3 (∃!x A x = B ↔ (x A x = B ∃*x A x = B))
64, 5mpbiran2 830 . 2 (∃!x A x = Bx A x = B)
71, 6bitr4i 176 1 (B A∃!x A x = B)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1223   ∈ wcel 1366  ∃*wmo 1874  ∃wrex 2276  ∃!wreu 2277  ∃*wrmo 2278 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-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-rex 2281  df-reu 2282  df-rmo 2283  df-v 2528 This theorem is referenced by:  divfnzn  8038
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