Theorem hbae-o 39534

Description: All variables are effectively bound in an identical variable specifier. Version of hbae 2465 using ax-c11 39518. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hbae-o	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)

Proof of Theorem hbae-o

Step	Hyp	Ref	Expression
1		ax-c5 39514	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2		ax-c9 39521	. . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
3	1, 2	syl7 75	. . . 4 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
4		ax-c11 39518	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
5	4	aecoms-o 39533	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
6		ax-c11 39518	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
7	6	pm2.43i 53	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
8		ax-c11 39518	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9	7, 8	syl5 35	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
10	9	aecoms-o 39533	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
11	3, 5, 10	pm2.61ii 185	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
12	11	axc4i-o 39529	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑧 𝑥 = 𝑦)
13		ax-11 2194	. 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
14	12, 13	syl 18	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-11 2194  ax-c5 39514  ax-c4 39515  ax-c7 39516  ax-c10 39517  ax-c11 39518  ax-c9 39521

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  dral1-o  39535  hbnae-o  39559  dral2-o  39561  aev-o  39562