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Mirrors > Home > MPE Home > Th. List > axc16gALT | Structured version Visualization version GIF version |
Description: Alternate proof of axc16g 2261 that uses df-sb 2070 and requires ax-10 2145, ax-11 2161, ax-13 2390. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc16gALT | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aev 2062 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥) | |
2 | axc16ALT 2528 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | |
3 | biidd 264 | . . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝜑 ↔ 𝜑)) | |
4 | 3 | dral1 2461 | . . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑)) |
5 | 4 | biimprd 250 | . 2 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝜑 → ∀𝑧𝜑)) |
6 | 1, 2, 5 | sylsyld 61 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 |
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: (None) |
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