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Theorem bj-hbaeb2 33717
 Description: Biconditional version of a form of hbae 2412 with commuted quantifiers, not requiring ax-11 2128. (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 2148 . . . . 5 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
2 axc9 2357 . . . . 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 2345 . . . 4 (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
5 axc11 2411 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
65pm2.43i 52 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
7 axc11r 2345 . . . . 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 184 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
109axc4i 2306 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑧 𝑥 = 𝑦)
11 sp 2148 . . 3 (∀𝑧 𝑥 = 𝑦𝑥 = 𝑦)
1211alimi 1797 . 2 (∀𝑥𝑧 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
1310, 12impbii 210 1 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥𝑧 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207  ∀wal 1523 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-10 2114  ax-12 2143  ax-13 2346 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770 This theorem is referenced by:  bj-hbaeb  33718  bj-dvv  33720
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