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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 2367. (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 2422 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) | |
2 | aecoms.1 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) | |
3 | 1, 2 | sylbi 216 | 1 ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-nf 1779 |
This theorem is referenced by: axc11 2425 nd4 10614 axrepnd 10618 axpownd 10625 axregnd 10628 axinfnd 10630 axacndlem5 10635 axacnd 10636 wl-ax11-lem1 37052 wl-ax11-lem3 37054 wl-ax11-lem9 37060 wl-ax11-lem10 37061 e2ebind 44002 |
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