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Theorem reueqd 2817
 Description: Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
Hypothesis
Ref Expression
raleqd.1 (A = B → (φψ))
Assertion
Ref Expression
reueqd (A = B → (∃!x A φ∃!x B ψ))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem reueqd
StepHypRef Expression
1 reueq1 2809 . 2 (A = B → (∃!x A φ∃!x B φ))
2 raleqd.1 . . 3 (A = B → (φψ))
32reubidv 2795 . 2 (A = B → (∃!x B φ∃!x B ψ))
41, 3bitrd 244 1 (A = B → (∃!x A φ∃!x B ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  ∃!wreu 2616 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-cleq 2346  df-clel 2349  df-nfc 2478  df-reu 2621 This theorem is referenced by: (None)
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