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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. (Contributed by NM, 10-May-1993.) |
Ref | Expression |
---|---|
aecoms.1 | ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) |
Ref | Expression |
---|---|
aecoms | ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aecom 2393 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) | |
2 | aecoms.1 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) | |
3 | 1, 2 | sylbi 209 | 1 ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-10 2135 ax-12 2163 ax-13 2334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1824 df-nf 1828 |
This theorem is referenced by: axc11 2396 nd4 9749 axrepnd 9753 axpownd 9760 axregnd 9763 axinfnd 9765 axacndlem5 9770 axacnd 9771 wl-ax11-lem1 33959 wl-ax11-lem3 33961 wl-ax11-lem9 33967 wl-ax11-lem10 33968 e2ebind 39733 |
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