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Mirrors > Home > MPE Home > Th. List > Mathboxes > equid1 | Structured version Visualization version GIF version |
Description: Proof of equid 2016 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 1914; see the proof of equid 2016. See equid1ALT 37843 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 37802 | . . . 4 ⊢ (∀𝑥(∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) → (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))) | |
2 | ax-c5 37801 | . . . . 5 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑥) | |
3 | ax-c9 37808 | . . . . 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 1800 | . . 3 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) |
6 | ax-c10 37804 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) → 𝑥 = 𝑥) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥) |
8 | ax-c7 37803 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥) | |
9 | 7, 8 | pm2.61i 182 | 1 ⊢ 𝑥 = 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-c5 37801 ax-c4 37802 ax-c7 37803 ax-c10 37804 ax-c9 37808 |
This theorem is referenced by: equcomi1 37818 |
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