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Mirrors > Home > MPE Home > Th. List > axc16ALT | Structured version Visualization version GIF version |
Description: Alternate proof of axc16 2262, shorter but requiring ax-10 2145, ax-11 2161, ax-13 2390 and using df-nf 1785 and df-sb 2070. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc16ALT | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ12 2253 | . 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
2 | ax-5 1911 | . . 3 ⊢ (𝜑 → ∀𝑧𝜑) | |
3 | 2 | hbsb3 2526 | . 2 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑) |
4 | 1, 3 | axc16i 2458 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: axc16gALT 2529 |
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