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Theorem eleq2dALT 2900
Description: Alternate proof of eleq2d 2899, shorter at the expense of requiring ax-12 2178. (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 2816 . . . . . 6 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2sylib 221 . . . . 5 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
4319.21bi 2189 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
54anbi2d 631 . . 3 (𝜑 → ((𝑥 = 𝐶𝑥𝐴) ↔ (𝑥 = 𝐶𝑥𝐵)))
65exbidv 1922 . 2 (𝜑 → (∃𝑥(𝑥 = 𝐶𝑥𝐴) ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐵)))
7 dfclel 2895 . 2 (𝐶𝐴 ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐴))
8 dfclel 2895 . 2 (𝐶𝐵 ↔ ∃𝑥(𝑥 = 𝐶𝑥𝐵))
96, 7, 83bitr4g 317 1 (𝜑 → (𝐶𝐴𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2815  df-clel 2894
This theorem is referenced by: (None)
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