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Mirrors > Home > ILE Home > Th. List > ax16 | GIF version |
Description: Theorem showing that ax-16 1692 is redundant if ax-17 1416 is included in the
axiom system. The important part of the proof is provided by aev 1690.
See ax16ALT 1736 for an alternate proof that does not require ax-10 1393 or ax-12 1399. This theorem should not be referenced in any proof. Instead, use ax-16 1692 below so that theorems needing ax-16 1692 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 1690 | . 2 ⊢ (∀x x = y → ∀z x = z) | |
2 | ax-17 1416 | . . . 4 ⊢ (φ → ∀zφ) | |
3 | sbequ12 1651 | . . . . 5 ⊢ (x = z → (φ ↔ [z / x]φ)) | |
4 | 3 | biimpcd 148 | . . . 4 ⊢ (φ → (x = z → [z / x]φ)) |
5 | 2, 4 | alimdh 1353 | . . 3 ⊢ (φ → (∀z x = z → ∀z[z / x]φ)) |
6 | 2 | hbsb3 1686 | . . . 4 ⊢ ([z / x]φ → ∀x[z / x]φ) |
7 | stdpc7 1650 | . . . 4 ⊢ (z = x → ([z / x]φ → φ)) | |
8 | 6, 2, 7 | cbv3h 1628 | . . 3 ⊢ (∀z[z / x]φ → ∀xφ) |
9 | 5, 8 | syl6com 31 | . 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 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-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 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: dveeq2 1693 dveeq2or 1694 a16g 1741 exists2 1994 |
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