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Theorem hbae-o 33006
Description: All variables are effectively bound in an identical variable specifier. Version of hbae 2299 using ax-c11 32990. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbae-o (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Proof of Theorem hbae-o
StepHypRef Expression
1 ax-c5 32986 . . . . 5 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
2 ax-c9 32993 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
31, 2syl7 71 . . . 4 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
4 ax-c11 32990 . . . . 5 (∀𝑥 𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
54aecoms-o 33005 . . . 4 (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
6 ax-c11 32990 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
76pm2.43i 49 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
8 ax-c11 32990 . . . . . 6 (∀𝑦 𝑦 = 𝑧 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
97, 8syl5 33 . . . . 5 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
109aecoms-o 33005 . . . 4 (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
113, 5, 10pm2.61ii 175 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
1211axc4i-o 33001 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑧 𝑥 = 𝑦)
13 ax-11 2020 . 2 (∀𝑥𝑧 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
1412, 13syl 17 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-11 2020  ax-c5 32986  ax-c4 32987  ax-c7 32988  ax-c10 32989  ax-c11 32990  ax-c9 32993
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  dral1-o  33007  hbnae-o  33031  dral2-o  33033  aev-o  33034
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