Theorem equid1 39391

Description: Proof of equid 2019 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 1917; see the proof of equid 2019. See equid1ALT 39417 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

Step	Hyp	Ref	Expression
1		ax-c4 39376	. . . 4 ⊢ (∀𝑥(∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) → (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)))
2		ax-c5 39375	. . . . 5 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑥)
3		ax-c9 39382	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑥 → (¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)))
4	2, 2, 3	sylc 65	. . . 4 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
5	1, 4	mpg 1804	. . 3 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
6		ax-c10 39378	. . 3 ⊢ (∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) → 𝑥 = 𝑥)
7	5, 6	syl 17	. 2 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥)
8		ax-c7 39377	. 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥)
9	7, 8	pm2.61i 183	1 ⊢ 𝑥 = 𝑥

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-c5 39375  ax-c4 39376  ax-c7 39377  ax-c10 39378  ax-c9 39382

This theorem is referenced by:  equcomi1  39392