Theorem hbae-o 34188

Description: All variables are effectively bound in an identical variable specifier. Version of hbae 2315 using ax-c11 34172. (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 34168	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2		ax-c9 34175	. . . . 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		ax-c11 34172	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
5	4	aecoms-o 34187	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
6		ax-c11 34172	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
7	6	pm2.43i 52	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
8		ax-c11 34172	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9	7, 8	syl5 34	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
10	9	aecoms-o 34187	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
11	3, 5, 10	pm2.61ii 177	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
12	11	axc4i-o 34183	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑧 𝑥 = 𝑦)
13		ax-11 2034	. 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
14	12, 13	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-11 2034  ax-c5 34168  ax-c4 34169  ax-c7 34170  ax-c10 34171  ax-c11 34172  ax-c9 34175

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  dral1-o  34189  hbnae-o  34213  dral2-o  34215  aev-o  34216