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Theorem eleq2dOLD 2674
 Description: Obsolete proof of eleq2d 2673 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 2604 . . . 4 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2sylib 207 . . 3 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
4 biimp 204 . . . . . 6 ((𝑥𝐴𝑥𝐵) → (𝑥𝐴𝑥𝐵))
54anim2d 587 . . . . 5 ((𝑥𝐴𝑥𝐵) → ((𝑥 = 𝐶𝑥𝐴) → (𝑥 = 𝐶𝑥𝐵)))
65aleximi 1749 . . . 4 (∀𝑥(𝑥𝐴𝑥𝐵) → (∃𝑥(𝑥 = 𝐶𝑥𝐴) → ∃𝑥(𝑥 = 𝐶𝑥𝐵)))
7 biimpr 209 . . . . . 6 ((𝑥𝐴𝑥𝐵) → (𝑥𝐵𝑥𝐴))
87anim2d 587 . . . . 5 ((𝑥𝐴𝑥𝐵) → ((𝑥 = 𝐶𝑥𝐵) → (𝑥 = 𝐶𝑥𝐴)))
98aleximi 1749 . . . 4 (∀𝑥(𝑥𝐴𝑥𝐵) → (∃𝑥(𝑥 = 𝐶𝑥𝐵) → ∃𝑥(𝑥 = 𝐶𝑥𝐴)))
106, 9impbid 201 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) → (∃𝑥(𝑥 = 𝐶𝑥𝐴) ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐵)))
113, 10syl 17 . 2 (𝜑 → (∃𝑥(𝑥 = 𝐶𝑥𝐴) ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐵)))
12 df-clel 2606 . 2 (𝐶𝐴 ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐴))
13 df-clel 2606 . 2 (𝐶𝐵 ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐵))
1411, 12, 133bitr4g 302 1 (𝜑 → (𝐶𝐴𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606 This theorem is referenced by: (None)
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