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Mirrors > Home > MPE Home > Th. List > axc16g | Structured version Visualization version GIF version |
Description: Generalization of axc16 2257. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 2015 }, Theorem ax12v 2176. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2374, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2261. (Revised by Wolf Lammen, 11-Oct-2021.) |
Ref | Expression |
---|---|
axc16g | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aevlem 2062 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑤) | |
2 | ax12v 2176 | . . . 4 ⊢ (𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤 → 𝜑))) | |
3 | 2 | sps 2182 | . . 3 ⊢ (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤 → 𝜑))) |
4 | pm2.27 42 | . . . 4 ⊢ (𝑧 = 𝑤 → ((𝑧 = 𝑤 → 𝜑) → 𝜑)) | |
5 | 4 | al2imi 1822 | . . 3 ⊢ (∀𝑧 𝑧 = 𝑤 → (∀𝑧(𝑧 = 𝑤 → 𝜑) → ∀𝑧𝜑)) |
6 | 3, 5 | syld 47 | . 2 ⊢ (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧𝜑)) |
7 | 1, 6 | syl 17 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) |
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 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 |
This theorem is referenced by: axc16 2257 axc16gb 2258 axc16nf 2259 axc11v 2260 axc16nfALT 2439 |
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