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Theorem eleq2w2ALT 34875
Description: Alternate proof of eleq2w2 2735 and special instance of eleq2 2822. (Contributed by BJ, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eleq2w2ALT (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))

Proof of Theorem eleq2w2ALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2732 . . 3 (𝐴 = 𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
21biimpi 219 . 2 (𝐴 = 𝐵 → ∀𝑦(𝑦𝐴𝑦𝐵))
3 eleq1w 2816 . . . 4 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
4 eleq1w 2816 . . . 4 (𝑦 = 𝑥 → (𝑦𝐵𝑥𝐵))
53, 4bibi12d 349 . . 3 (𝑦 = 𝑥 → ((𝑦𝐴𝑦𝐵) ↔ (𝑥𝐴𝑥𝐵)))
65spvv 2008 . 2 (∀𝑦(𝑦𝐴𝑦𝐵) → (𝑥𝐴𝑥𝐵))
72, 6syl 17 1 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1540   = wceq 1542  wcel 2114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-cleq 2731  df-clel 2812
This theorem is referenced by: (None)
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