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Mirrors > Home > ILE Home > Th. List > ax16ALT | GIF version |
Description: Version of ax16 1691 that doesn't require ax-10 1393 or ax-12 1399 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax16ALT | ⊢ (∀x x = y → (φ → ∀xφ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ12 1651 | . 2 ⊢ (x = z → (φ ↔ [z / x]φ)) | |
2 | ax-17 1416 | . . 3 ⊢ (φ → ∀zφ) | |
3 | 2 | hbsb3 1686 | . 2 ⊢ ([z / x]φ → ∀x[z / x]φ) |
4 | 1, 3 | ax16i 1735 | 1 ⊢ (∀x x = y → (φ → ∀xφ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1240 [wsb 1642 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 |
This theorem is referenced by: dvelimALT 1883 dvelimfv 1884 |
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