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Mirrors > Home > MPE Home > Th. List > axc16ALT | Structured version Visualization version GIF version |
Description: Alternate proof of axc16 2244, shorter but requiring ax-10 2129, ax-11 2146, ax-13 2365 and using df-nf 1778 and df-sb 2060. (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 2235 | . 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
2 | ax-5 1905 | . . 3 ⊢ (𝜑 → ∀𝑧𝜑) | |
3 | 2 | hbsb3 2480 | . 2 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑) |
4 | 1, 3 | axc16i 2429 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 [wsb 2059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-11 2146 ax-12 2163 ax-13 2365 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1774 df-nf 1778 df-sb 2060 |
This theorem is referenced by: axc16gALT 2483 |
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