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Theorem rmo2ilem 2841
 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 1356 . . . 4 (x((x A φ) → x = y) ↔ x(x A → (φx = y)))
3 df-ral 2305 . . . 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 1493 . 2 (yx((x A φ) → x = y) ↔ yx A (φx = y))
6 nfv 1418 . . . . 5 y x A
7 rmo2.1 . . . . 5 yφ
86, 7nfan 1454 . . . 4 y(x A φ)
98mo2r 1949 . . 3 (yx((x A φ) → x = y) → ∃*x(x A φ))
10 df-rmo 2308 . . 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 1240   = wceq 1242  Ⅎwnf 1346  ∃wex 1378   ∈ wcel 1390  ∃*wmo 1898  ∀wral 2300  ∃*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 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-ral 2305  df-rmo 2308 This theorem is referenced by:  rmo2i  2842
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