![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2375. (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 2430 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) | |
2 | aecoms.1 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) | |
3 | 1, 2 | sylbi 217 | 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 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 |
This theorem is referenced by: axc11 2433 nd4 10628 axrepnd 10632 axpownd 10639 axregnd 10642 axinfnd 10644 axacndlem5 10649 axacnd 10650 wl-ax11-lem1 37566 wl-ax11-lem3 37568 wl-ax11-lem9 37574 wl-ax11-lem10 37575 e2ebind 44561 |
Copyright terms: Public domain | W3C validator |