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Mirrors > Home > ILE Home > Th. List > a16gb | GIF version |
Description: A generalization of axiom ax-16 1695. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
a16gb | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | a16g 1744 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | |
2 | ax-4 1400 | . 2 ⊢ (∀𝑧𝜑 → 𝜑) | |
3 | 1, 2 | impbid1 130 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 |
This theorem is referenced by: (None) |
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