Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > aev | Structured version Visualization version GIF version |
Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2162. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2373, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2179. (Revised by Wolf Lammen, 19-Mar-2021.) |
Ref | Expression |
---|---|
aev | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aevlem 2065 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑣 𝑣 = 𝑤) | |
2 | aeveq 2066 | . . 3 ⊢ (∀𝑣 𝑣 = 𝑤 → 𝑡 = 𝑢) | |
3 | 2 | alrimiv 1934 | . 2 ⊢ (∀𝑣 𝑣 = 𝑤 → ∀𝑧 𝑡 = 𝑢) |
4 | 1, 3 | syl 17 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 |
This theorem is referenced by: aev2 2068 naev 2070 axc11n 2427 axc16gALT 2495 aevdemo 28409 axc11n11r 34520 wl-ax11-lem2 35392 |
Copyright terms: Public domain | W3C validator |