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Theorem eleq2dALT 2837
Description: Alternate proof of eleq2d 2836, shorter at the expense of requiring ax-12 2203. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 20-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
eleq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
eleq2dALT (𝜑 → (𝐶𝐴𝐶𝐵))

Proof of Theorem eleq2dALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1d.1 . . . . . 6 (𝜑𝐴 = 𝐵)
2 dfcleq 2765 . . . . . 6 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2sylib 208 . . . . 5 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
4319.21bi 2213 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
54anbi2d 614 . . 3 (𝜑 → ((𝑥 = 𝐶𝑥𝐴) ↔ (𝑥 = 𝐶𝑥𝐵)))
65exbidv 2002 . 2 (𝜑 → (∃𝑥(𝑥 = 𝐶𝑥𝐴) ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐵)))
7 df-clel 2767 . 2 (𝐶𝐴 ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐴))
8 df-clel 2767 . 2 (𝐶𝐵 ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐵))
96, 7, 83bitr4g 303 1 (𝜑 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wex 1852  wcel 2145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-cleq 2764  df-clel 2767
This theorem is referenced by: (None)
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