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Theorem reusv1 4156
Description: Two ways to express single-valuedness of a class expression 𝐶(y). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
reusv1 (y B φ → (∃!x A y B (φx = 𝐶) ↔ x A y B (φx = 𝐶)))
Distinct variable groups:   x,A   x,B   x,𝐶   φ,x   x,y
Allowed substitution hints:   φ(y)   A(y)   B(y)   𝐶(y)

Proof of Theorem reusv1
StepHypRef Expression
1 nfra1 2349 . . . 4 yy B (φx = 𝐶)
21nfmo 1917 . . 3 y∃*xy B (φx = 𝐶)
3 rsp 2363 . . . . . . . 8 (y B (φx = 𝐶) → (y B → (φx = 𝐶)))
43impd 242 . . . . . . 7 (y B (φx = 𝐶) → ((y B φ) → x = 𝐶))
54com12 27 . . . . . 6 ((y B φ) → (y B (φx = 𝐶) → x = 𝐶))
65alrimiv 1751 . . . . 5 ((y B φ) → x(y B (φx = 𝐶) → x = 𝐶))
7 moeq 2710 . . . . 5 ∃*x x = 𝐶
8 moim 1961 . . . . 5 (x(y B (φx = 𝐶) → x = 𝐶) → (∃*x x = 𝐶∃*xy B (φx = 𝐶)))
96, 7, 8mpisyl 1332 . . . 4 ((y B φ) → ∃*xy B (φx = 𝐶))
109ex 108 . . 3 (y B → (φ∃*xy B (φx = 𝐶)))
112, 10rexlimi 2420 . 2 (y B φ∃*xy B (φx = 𝐶))
12 mormo 2515 . 2 (∃*xy B (φx = 𝐶) → ∃*x A y B (φx = 𝐶))
13 reu5 2516 . . 3 (∃!x A y B (φx = 𝐶) ↔ (x A y B (φx = 𝐶) ∃*x A y B (φx = 𝐶)))
1413rbaib 829 . 2 (∃*x A y B (φx = 𝐶) → (∃!x A y B (φx = 𝐶) ↔ x A y B (φx = 𝐶)))
1511, 12, 143syl 17 1 (y B φ → (∃!x A y B (φx = 𝐶) ↔ x A y B (φx = 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  ∃*wmo 1898  wral 2300  wrex 2301  ∃!wreu 2302  ∃*wrmo 2303
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308  df-v 2553
This theorem is referenced by: (None)
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