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Theorem reueq1OLD 3411
Description: Obsolete version of reueq1 3409 as of 12-Mar-2025. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2129, ax-11 2146, and ax-12 2163. (Revised by Steven Nguyen, 30-Apr-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
reueq1OLD (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reueq1OLD
StepHypRef Expression
1 eleq2 2816 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21anbi1d 629 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜑)))
32eubidv 2574 . 2 (𝐴 = 𝐵 → (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑥(𝑥𝐵𝜑)))
4 df-reu 3371 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
5 df-reu 3371 . 2 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
63, 4, 53bitr4g 314 1 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  ∃!weu 2556  ∃!wreu 3368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-mo 2528  df-eu 2557  df-cleq 2718  df-clel 2804  df-reu 3371
This theorem is referenced by: (None)
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