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Mirrors > Home > MPE Home > Th. List > Mathboxes > axc11nfromc11 | Structured version Visualization version GIF version |
Description: Rederivation of ax-c11n 33191 from original version ax-c11 33190. See theorem
axc11 2302 for the derivation of ax-c11 33190 from ax-c11n 33191.
This theorem should not be referenced in any proof. Instead, use ax-c11n 33191 above so that uses of ax-c11n 33191 can be more easily identified, or use aecom-o 33204 when this form is needed for studies involving ax-c11 33190 and omitting ax-5 1827. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc11nfromc11 | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-c11 33190 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)) | |
2 | 1 | pm2.43i 50 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦) |
3 | equcomi 1931 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | |
4 | 3 | alimi 1730 | . 2 ⊢ (∀𝑦 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
5 | 2, 4 | syl 17 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-c11 33190 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: (None) |
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