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Theorem rmoan 3034
 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 2255 . . 3 (∃*x(x A φ) → ∃*x(ψ (x A φ)))
2 an12 772 . . . 4 ((ψ (x A φ)) ↔ (x A (ψ φ)))
32mobii 2240 . . 3 (∃*x(ψ (x A φ)) ↔ ∃*x(x A (ψ φ)))
41, 3sylib 188 . 2 (∃*x(x A φ) → ∃*x(x A (ψ φ)))
5 df-rmo 2622 . 2 (∃*x A φ∃*x(x A φ))
6 df-rmo 2622 . 2 (∃*x A (ψ φ) ↔ ∃*x(x A (ψ φ)))
74, 5, 63imtr4i 257 1 (∃*x A φ∃*x A (ψ φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∃*wmo 2205  ∃*wrmo 2617 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-rmo 2622 This theorem is referenced by: (None)
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