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Mirrors > Home > ILE Home > Th. List > ax16 | GIF version |
Description: Theorem showing that ax-16 1770 is redundant if ax-17 1491 is included in the
axiom system. The important part of the proof is provided by aev 1768.
See ax16ALT 1815 for an alternate proof that does not require ax-10 1468 or ax-12 1474. This theorem should not be referenced in any proof. Instead, use ax-16 1770 below so that theorems needing ax-16 1770 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
Ref | Expression |
---|---|
ax16 | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aev 1768 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑧) | |
2 | ax-17 1491 | . . . 4 ⊢ (𝜑 → ∀𝑧𝜑) | |
3 | sbequ12 1729 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
4 | 3 | biimpcd 158 | . . . 4 ⊢ (𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑥]𝜑)) |
5 | 2, 4 | alimdh 1428 | . . 3 ⊢ (𝜑 → (∀𝑧 𝑥 = 𝑧 → ∀𝑧[𝑧 / 𝑥]𝜑)) |
6 | 2 | hbsb3 1764 | . . . 4 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑) |
7 | stdpc7 1728 | . . . 4 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑 → 𝜑)) | |
8 | 6, 2, 7 | cbv3h 1706 | . . 3 ⊢ (∀𝑧[𝑧 / 𝑥]𝜑 → ∀𝑥𝜑) |
9 | 5, 8 | syl6com 35 | . 2 ⊢ (∀𝑧 𝑥 = 𝑧 → (𝜑 → ∀𝑥𝜑)) |
10 | 1, 9 | syl 14 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1314 [wsb 1720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 |
This theorem is referenced by: dveeq2 1771 dveeq2or 1772 a16g 1820 exists2 2074 |
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