Theorem bj-hbaeb2 34738

Description: Biconditional version of a form of hbae 2430 with commuted quantifiers, not requiring ax-11 2158. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-hbaeb2	⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦)

Proof of Theorem bj-hbaeb2

Step	Hyp	Ref	Expression
1		sp 2180	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2		axc9 2381	. . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
3	1, 2	syl7 74	. . . 4 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
4		axc11r 2367	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
5		axc11 2429	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
6	5	pm2.43i 52	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
7		axc11r 2367	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
8	6, 7	syl5 34	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9	3, 4, 8	pm2.61ii 186	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
10	9	axc4i 2321	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑧 𝑥 = 𝑦)
11		sp 2180	. . 3 ⊢ (∀𝑧 𝑥 = 𝑦 → 𝑥 = 𝑦)
12	11	alimi 1819	. 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
13	10, 12	impbii 212	1 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-12 2175  ax-13 2371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792

This theorem is referenced by:  bj-hbaeb  34739  bj-dvv  34741