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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 2021. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2234, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2034. (Revised by Wolf Lammen, 19-Mar-2021.) |
Ref | Expression |
---|---|
aev | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aevlem 1968 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑣 𝑣 = 𝑤) | |
2 | aeveq 1969 | . . 3 ⊢ (∀𝑣 𝑣 = 𝑤 → 𝑡 = 𝑢) | |
3 | 2 | alrimiv 1842 | . 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 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: aev2 1973 aev2ALT 1974 axc16nfOLD 2149 axc11n 2295 axc11nOLD 2296 axc16gALT 2355 aevdemo 26709 axc11n11r 31860 wl-naev 32481 wl-hbnaev 32484 wl-ax11-lem2 32542 |
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