![]() |
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 2128. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2346, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2143. (Revised by Wolf Lammen, 19-Mar-2021.) |
Ref | Expression |
---|---|
aev | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aevlem 2035 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑣 𝑣 = 𝑤) | |
2 | aeveq 2036 | . . 3 ⊢ (∀𝑣 𝑣 = 𝑤 → 𝑡 = 𝑢) | |
3 | 2 | alrimiv 1909 | . 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 1523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1766 |
This theorem is referenced by: aev2 2038 naev 2040 axc11n 2407 axc16gALT 2485 aevdemo 27927 axc11n11r 33621 wl-ax11-lem2 34370 |
Copyright terms: Public domain | W3C validator |