Theorem bj-hbaeb2 31993

Description: Biconditional version of a form of hbae 2303 with commuted quantifiers, not requiring ax-11 2021. (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 2041	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2		axc9 2290	. . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
3	1, 2	syl7 72	. . . 4 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
4		axc11r 2175	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
5		axc11 2302	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
6	5	pm2.43i 50	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
7		axc11r 2175	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
8	6, 7	syl5 33	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9	3, 4, 8	pm2.61ii 176	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
10	9	axc4i 2116	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑧 𝑥 = 𝑦)
11		sp 2041	. . 3 ⊢ (∀𝑧 𝑥 = 𝑦 → 𝑥 = 𝑦)
12	11	alimi 1730	. 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
13	10, 12	impbii 198	1 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀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  ax-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by:  bj-hbaeb  31994  bj-dvv  31996