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Theorem aevlemOLD 1976
 Description: Old proof of aevlem 1968. Obsolete as of 29-Mar-2021. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2234, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
aevlemOLD (∀𝑧 𝑧 = 𝑤 → ∀𝑦 𝑦 = 𝑥)
Distinct variable groups:   𝑧,𝑤   𝑥,𝑦

Proof of Theorem aevlemOLD
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 cbvaev 1966 . 2 (∀𝑧 𝑧 = 𝑤 → ∀𝑣 𝑣 = 𝑤)
2 ax5d 1828 . . 3 (¬ ∀𝑧 𝑧 = 𝑣 → (𝑣 = 𝑤 → ∀𝑧 𝑣 = 𝑤))
32axc11nlemOLD2 1975 . 2 (∀𝑣 𝑣 = 𝑤 → ∀𝑧 𝑧 = 𝑣)
4 cbvaev 1966 . 2 (∀𝑧 𝑧 = 𝑣 → ∀𝑥 𝑥 = 𝑣)
5 ax5d 1828 . . 3 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑣 → ∀𝑦 𝑥 = 𝑣))
65axc11nlemOLD2 1975 . 2 (∀𝑥 𝑥 = 𝑣 → ∀𝑦 𝑦 = 𝑥)
71, 3, 4, 64syl 19 1 (∀𝑧 𝑧 = 𝑤 → ∀𝑦 𝑦 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473 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-5 1827  ax-6 1875  ax-7 1922 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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