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Theorem reueqd 3287
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 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reueqd
StepHypRef Expression
1 reueq1 3279 . 2 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
2 raleqd.1 . . 3 (𝐴 = 𝐵 → (𝜑𝜓))
32reubidv 3265 . 2 (𝐴 = 𝐵 → (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥𝐵 𝜓))
41, 3bitrd 268 1 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  ∃!wreu 3052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-eu 2611  df-cleq 2753  df-clel 2756  df-nfc 2891  df-reu 3057
This theorem is referenced by:  aceq1  9130
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