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Mirrors > Home > MPE Home > Th. List > axc16g | Structured version Visualization version GIF version |
Description: Generalization of axc16 2120. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 1922 }, theorem ax12v 2035. (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 2234, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2124. (Revised by Wolf Lammen, 11-Oct-2021.) |
Ref | Expression |
---|---|
axc16g | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aevlem 1968 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑤) | |
2 | ax12v2 2036 | . . . 4 ⊢ (𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤 → 𝜑))) | |
3 | 2 | sps 2043 | . . 3 ⊢ (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧(𝑧 = 𝑤 → 𝜑))) |
4 | pm2.27 41 | . . . 4 ⊢ (𝑧 = 𝑤 → ((𝑧 = 𝑤 → 𝜑) → 𝜑)) | |
5 | 4 | al2imi 1733 | . . 3 ⊢ (∀𝑧 𝑧 = 𝑤 → (∀𝑧(𝑧 = 𝑤 → 𝜑) → ∀𝑧𝜑)) |
6 | 3, 5 | syld 46 | . 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 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 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: axc16 2120 axc16gb 2121 axc16nf 2122 axc11v 2123 axc11rv 2124 aevOLD 2148 axc16nfALT 2311 |
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