Theorem hbae-o 2153

Description: All variables are effectively bound in an identical variable specifier. Version of hbae 1953 using ax-10o 2139. (Contributed by NM, 5-Aug-1993.) (Proof modification is disccouraged.) (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-4 2135	. . . . 5 ⊢ (∀x x = y → x = y)
2		ax-12o 2142	. . . . 5 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))
3	1, 2	syl7 63	. . . 4 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (∀x x = y → ∀z x = y)))
4		ax-10o 2139	. . . . 5 ⊢ (∀x x = z → (∀x x = y → ∀z x = y))
5	4	aecoms-o 2152	. . . 4 ⊢ (∀z z = x → (∀x x = y → ∀z x = y))
6		ax-10o 2139	. . . . . . 7 ⊢ (∀x x = y → (∀x x = y → ∀y x = y))
7	6	pm2.43i 43	. . . . . 6 ⊢ (∀x x = y → ∀y x = y)
8		ax-10o 2139	. . . . . 6 ⊢ (∀y y = z → (∀y x = y → ∀z x = y))
9	7, 8	syl5 28	. . . . 5 ⊢ (∀y y = z → (∀x x = y → ∀z x = y))
10	9	aecoms-o 2152	. . . 4 ⊢ (∀z z = y → (∀x x = y → ∀z x = y))
11	3, 5, 10	pm2.61ii 157	. . 3 ⊢ (∀x x = y → ∀z x = y)
12	11	a5i-o 2150	. 2 ⊢ (∀x x = y → ∀x∀z x = y)
13		ax-7 1734	. 2 ⊢ (∀x∀z x = y → ∀z∀x x = y)
14	12, 13	syl 15	1 ⊢ (∀x x = y → ∀z∀x x = y)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-7 1734  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  dral1-o  2154  hbnae-o  2179  dral2-o  2181  aev-o  2182