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Mirrors > Home > MPE Home > Th. List > aevlem | Structured version Visualization version GIF version |
Description: Lemma for aev 2055 and axc16g 2258. Change free and bound variables. Instance of aev 2055. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2375, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Reduce axiom usage. (Revised by BJ, 29-Mar-2021.) |
Ref | Expression |
---|---|
aevlem | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvaev 2051 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑦) | |
2 | aevlem0 2052 | . 2 ⊢ (∀𝑢 𝑢 = 𝑦 → ∀𝑥 𝑥 = 𝑢) | |
3 | cbvaev 2051 | . 2 ⊢ (∀𝑥 𝑥 = 𝑢 → ∀𝑡 𝑡 = 𝑢) | |
4 | aevlem0 2052 | . 2 ⊢ (∀𝑡 𝑡 = 𝑢 → ∀𝑧 𝑧 = 𝑡) | |
5 | 1, 2, 3, 4 | 4syl 19 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 |
This theorem is referenced by: aeveq 2054 aev 2055 axc16g 2258 bj-axc16g16 36667 bj-axc11nv 36790 bj-aecomsv 36791 |
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