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Theorem equid1 35915
Description: Proof of equid 2010 from our older axioms. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and requires no dummy variables. A simpler proof, similar to Tarski's, is possible if we make use of ax-5 1902; see the proof of equid 2010. See equid1ALT 35941 for an alternate proof. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
equid1 𝑥 = 𝑥

Proof of Theorem equid1
StepHypRef Expression
1 ax-c4 35900 . . . 4 (∀𝑥(∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) → (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)))
2 ax-c5 35899 . . . . 5 (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑥)
3 ax-c9 35906 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑥 → (¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)))
42, 2, 3sylc 65 . . . 4 (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
51, 4mpg 1789 . . 3 (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
6 ax-c10 35902 . . 3 (∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) → 𝑥 = 𝑥)
75, 6syl 17 . 2 (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥𝑥 = 𝑥)
8 ax-c7 35901 . 2 (¬ ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥𝑥 = 𝑥)
97, 8pm2.61i 183 1 𝑥 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-c5 35899  ax-c4 35900  ax-c7 35901  ax-c10 35902  ax-c9 35906
This theorem is referenced by:  equcomi1  35916
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