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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aecom-o | Structured version Visualization version GIF version |
Description: Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2418 using ax-c11 38213. Unlike axc11nfromc11 38252, this version does not require ax-5 1905 (see comment of equcomi1 38226). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
aecom-o | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-c11 38213 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)) | |
2 | 1 | pm2.43i 52 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦) |
3 | equcomi1 38226 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | |
4 | 3 | alimi 1805 | . 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 1531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-c5 38209 ax-c4 38210 ax-c7 38211 ax-c10 38212 ax-c11 38213 ax-c9 38216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 |
This theorem is referenced by: aecoms-o 38228 naecoms-o 38253 aev-o 38257 ax12indalem 38271 |
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