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Theorem equid1 33661
Description: Proof of equid 1936 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 1836; see the proof of equid 1936. See equid1ALT 33687 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 33646 . . . 4 (∀𝑥(∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) → (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)))
2 ax-c5 33645 . . . . 5 (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑥)
3 ax-c9 33652 . . . . 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 1721 . . 3 (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
6 ax-c10 33648 . . 3 (∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) → 𝑥 = 𝑥)
75, 6syl 17 . 2 (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥𝑥 = 𝑥)
8 ax-c7 33647 . 2 (¬ ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥𝑥 = 𝑥)
97, 8pm2.61i 176 1 𝑥 = 𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-c5 33645  ax-c4 33646  ax-c7 33647  ax-c10 33648  ax-c9 33652
This theorem is referenced by:  equcomi1  33662
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