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Theorem eleq2dOLD 2669
Description: Obsolete proof of eleq2d 2668 as of 16-Nov-2020. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
eleq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
eleq2dOLD (𝜑 → (𝐶𝐴𝐶𝐵))

Proof of Theorem eleq2dOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1d.1 . . . 4 (𝜑𝐴 = 𝐵)
2 dfcleq 2599 . . . 4 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2sylib 206 . . 3 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
4 biimp 203 . . . . . 6 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴𝑥𝐵))
54anim2d 586 . . . . 5 ((𝑥𝐴𝑥𝐵) → ((𝑥 = 𝐶𝑥𝐴) → (𝑥 = 𝐶𝑥𝐵)))
65aleximi 1747 . . . 4 (∀𝑥(𝑥𝐴𝑥𝐵) → (∃𝑥(𝑥 = 𝐶𝑥𝐴) → ∃𝑥(𝑥 = 𝐶𝑥𝐵)))
7 biimpr 208 . . . . . 6 ((𝑥𝐴𝑥𝐵) → (𝑥𝐵𝑥𝐴))
87anim2d 586 . . . . 5 ((𝑥𝐴𝑥𝐵) → ((𝑥 = 𝐶𝑥𝐵) → (𝑥 = 𝐶𝑥𝐴)))
98aleximi 1747 . . . 4 (∀𝑥(𝑥𝐴𝑥𝐵) → (∃𝑥(𝑥 = 𝐶𝑥𝐵) → ∃𝑥(𝑥 = 𝐶𝑥𝐴)))
106, 9impbid 200 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) → (∃𝑥(𝑥 = 𝐶𝑥𝐴) ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐵)))
113, 10syl 17 . 2 (𝜑 → (∃𝑥(𝑥 = 𝐶𝑥𝐴) ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐵)))
12 df-clel 2601 . 2 (𝐶𝐴 ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐴))
13 df-clel 2601 . 2 (𝐶𝐵 ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐵))
1411, 12, 133bitr4g 301 1 (𝜑 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-cleq 2598  df-clel 2601
This theorem is referenced by: (None)
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