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Theorem aevALTOLD 2320
Description: Older alternate proof of aev 1980. Obsolete as of 30-Mar-2021. (Contributed by NM, 8-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
aevALTOLD (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
Distinct variable group:   𝑥,𝑦

Proof of Theorem aevALTOLD
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 hbae 2314 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
2 aevlem 1978 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑣)
3 ax7 1940 . . . 4 (𝑢 = 𝑤 → (𝑢 = 𝑣𝑤 = 𝑣))
43spimv 2256 . . 3 (∀𝑢 𝑢 = 𝑣𝑤 = 𝑣)
52, 4syl 17 . 2 (∀𝑥 𝑥 = 𝑦𝑤 = 𝑣)
61, 5alrimih 1748 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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