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Theorem rmoan 2733
Description: Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
Assertion
Ref Expression
rmoan (∃*x A φ∃*x A (ψ φ))

Proof of Theorem rmoan
StepHypRef Expression
1 moan 1966 . . 3 (∃*x(x A φ) → ∃*x(ψ (x A φ)))
2 an12 495 . . . 4 ((ψ (x A φ)) ↔ (x A (ψ φ)))
32mobii 1934 . . 3 (∃*x(ψ (x A φ)) ↔ ∃*x(x A (ψ φ)))
41, 3sylib 127 . 2 (∃*x(x A φ) → ∃*x(x A (ψ φ)))
5 df-rmo 2308 . 2 (∃*x A φ∃*x(x A φ))
6 df-rmo 2308 . 2 (∃*x A (ψ φ) ↔ ∃*x(x A (ψ φ)))
74, 5, 63imtr4i 190 1 (∃*x A φ∃*x A (ψ φ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  ∃*wmo 1898  ∃*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-bndl 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-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-rmo 2308
This theorem is referenced by: (None)
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