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Theorem reuun1 3537
 Description: Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
Assertion
Ref Expression
reuun1 ((x A φ ∃!x (AB)(φ ψ)) → ∃!x A φ)
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem reuun1
StepHypRef Expression
1 ssun1 3426 . 2 A (AB)
2 orc 374 . . 3 (φ → (φ ψ))
32rgenw 2681 . 2 x A (φ → (φ ψ))
4 reuss2 3535 . 2 (((A (AB) x A (φ → (φ ψ))) (x A φ ∃!x (AB)(φ ψ))) → ∃!x A φ)
51, 3, 4mpanl12 663 1 ((x A φ ∃!x (AB)(φ ψ)) → ∃!x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358  ∀wral 2614  ∃wrex 2615  ∃!wreu 2616   ∪ cun 3207   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-reu 2621  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259 This theorem is referenced by: (None)
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