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Theorem equid1ALT 33026
Description: Alternate proof of equid 1924 and equid1 33000 from older axioms ax-c7 32986, ax-c10 32987 and ax-c9 32991. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equid1ALT 𝑥 = 𝑥

Proof of Theorem equid1ALT
StepHypRef Expression
1 ax-c9 32991 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑥 → (¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)))
21pm2.43i 49 . . . 4 (¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
32alimi 1728 . . 3 (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
4 ax-c10 32987 . . 3 (∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) → 𝑥 = 𝑥)
53, 4syl 17 . 2 (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥𝑥 = 𝑥)
6 ax-c7 32986 . 2 (¬ ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥𝑥 = 𝑥)
75, 6pm2.61i 174 1 𝑥 = 𝑥
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 1711  ax-4 1726  ax-c7 32986  ax-c10 32987  ax-c9 32991
This theorem is referenced by: (None)
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