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Mirrors > Home > MPE Home > Th. List > Mathboxes > equid1 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
equid1 | ⊢ 𝑥 = 𝑥 |
Step | Hyp | Ref | 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))) | |
4 | 2, 2, 3 | sylc 65 | . . . 4 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) |
5 | 1, 4 | mpg 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) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥) |
8 | ax-c7 35901 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥) | |
9 | 7, 8 | pm2.61i 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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