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Theorem bj-hbaeb2 33140
Description: Biconditional version of a form of hbae 2467 with commuted quantifiers, not requiring ax-11 2190. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-hbaeb2 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑧 𝑥 = 𝑦)

Proof of Theorem bj-hbaeb2
StepHypRef Expression
1 sp 2207 . . . . 5 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
2 axc9 2458 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
31, 2syl7 74 . . . 4 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
4 axc11r 2349 . . . 4 (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
5 axc11 2466 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
65pm2.43i 52 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
7 axc11r 2349 . . . . 5 (∀𝑧 𝑧 = 𝑦 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
86, 7syl5 34 . . . 4 (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
93, 4, 8pm2.61ii 177 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
109axc4i 2295 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑧 𝑥 = 𝑦)
11 sp 2207 . . 3 (∀𝑧 𝑥 = 𝑦𝑥 = 𝑦)
1211alimi 1887 . 2 (∀𝑥𝑧 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
1310, 12impbii 199 1 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑧 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1629
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-10 2174  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858
This theorem is referenced by:  bj-hbaeb  33141  bj-dvv  33143
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