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Mirrors > Home > MPE Home > Th. List > aecoms | Structured version Visualization version GIF version |
Description: A commutation rule for identical variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by NM, 10-May-1993.) (New usage is discouraged.) |
Ref | Expression |
---|---|
aecoms.1 | ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) |
Ref | Expression |
---|---|
aecoms | ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aecom 2449 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) | |
2 | aecoms.1 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) | |
3 | 1, 2 | sylbi 219 | 1 ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀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: axc11 2452 nd4 10014 axrepnd 10018 axpownd 10025 axregnd 10028 axinfnd 10030 axacndlem5 10035 axacnd 10036 wl-ax11-lem1 34819 wl-ax11-lem3 34821 wl-ax11-lem9 34827 wl-ax11-lem10 34828 e2ebind 40904 |
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