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Theorem rmo2ilem 2824
Description: Condition implying restricted "at most one." (Contributed by Jim Kingdon, 14-Jul-2018.)
Hypothesis
Ref Expression
rmo2.1 yφ
Assertion
Ref Expression
rmo2ilem (yx A (φx = y) → ∃*x A φ)
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)

Proof of Theorem rmo2ilem
StepHypRef Expression
1 impexp 250 . . . . 5 (((x A φ) → x = y) ↔ (x A → (φx = y)))
21albii 1339 . . . 4 (x((x A φ) → x = y) ↔ x(x A → (φx = y)))
3 df-ral 2289 . . . 4 (x A (φx = y) ↔ x(x A → (φx = y)))
42, 3bitr4i 176 . . 3 (x((x A φ) → x = y) ↔ x A (φx = y))
54exbii 1478 . 2 (yx((x A φ) → x = y) ↔ yx A (φx = y))
6 nfv 1402 . . . . 5 y x A
7 rmo2.1 . . . . 5 yφ
86, 7nfan 1439 . . . 4 y(x A φ)
98mo2r 1934 . . 3 (yx((x A φ) → x = y) → ∃*x(x A φ))
10 df-rmo 2292 . . 3 (∃*x A φ∃*x(x A φ))
119, 10sylibr 137 . 2 (yx((x A φ) → x = y) → ∃*x A φ)
125, 11sylbir 125 1 (yx A (φx = y) → ∃*x A φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1226   = wceq 1228  wnf 1329  wex 1362   wcel 1374  ∃*wmo 1883  wral 2284  ∃*wrmo 2287
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-ral 2289  df-rmo 2292
This theorem is referenced by:  rmo2i  2825
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