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Mirrors > Home > MPE Home > Th. List > aecom | Structured version Visualization version GIF version |
Description: Commutation law for identical variable specifiers. Both sides of the biconditional are true when 𝑥 and 𝑦 are substituted with the same variable. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by NM, 10-May-1993.) Change to a biconditional. (Revised by BJ, 26-Sep-2019.) (New usage is discouraged.) |
Ref | Expression |
---|---|
aecom | ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axc11n 2448 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | |
2 | axc11n 2448 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦) | |
3 | 1, 2 | impbii 211 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 |
This theorem is referenced by: aecoms 2450 naecoms 2451 wl-nfae1 34782 |
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