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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 2371. (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 2426 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) | |
2 | aecoms.1 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) | |
3 | 1, 2 | sylbi 220 | 1 ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-10 2143 ax-12 2177 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 |
This theorem is referenced by: axc11 2429 nd4 10169 axrepnd 10173 axpownd 10180 axregnd 10183 axinfnd 10185 axacndlem5 10190 axacnd 10191 wl-ax11-lem1 35422 wl-ax11-lem3 35424 wl-ax11-lem9 35430 wl-ax11-lem10 35431 e2ebind 41797 |
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