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| Mirrors > Home > MPE Home > Th. List > aevlem | Structured version Visualization version GIF version | ||
| Description: Lemma for aev 2057 and axc16g 2260. Change free and bound variables. Instance of aev 2057. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2377, 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 2053 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑦) | |
| 2 | aevlem0 2054 | . 2 ⊢ (∀𝑢 𝑢 = 𝑦 → ∀𝑥 𝑥 = 𝑢) | |
| 3 | cbvaev 2053 | . 2 ⊢ (∀𝑥 𝑥 = 𝑢 → ∀𝑡 𝑡 = 𝑢) | |
| 4 | aevlem0 2054 | . 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 1538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 |
| This theorem is referenced by: aeveq 2056 aev 2057 axc16g 2260 bj-axc16g16 36685 bj-axc11nv 36808 bj-aecomsv 36809 |
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