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Theorem reueq1f 2663
Description: Equality theorem for restricted unique existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleq1f.1 𝑥𝐴
raleq1f.2 𝑥𝐵
Assertion
Ref Expression
reueq1f (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))

Proof of Theorem reueq1f
StepHypRef Expression
1 raleq1f.1 . . . 4 𝑥𝐴
2 raleq1f.2 . . . 4 𝑥𝐵
31, 2nfeq 2320 . . 3 𝑥 𝐴 = 𝐵
4 eleq2 2234 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 462 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
63, 5eubid 2026 . 2 (𝐴 = 𝐵 → (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐵𝜑)))
7 df-reu 2455 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
8 df-reu 2455 . 2 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
96, 7, 83bitr4g 222 1 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  ∃!weu 2019  wcel 2141  wnfc 2299  ∃!wreu 2450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-cleq 2163  df-clel 2166  df-nfc 2301  df-reu 2455
This theorem is referenced by:  reueq1  2667
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