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Mirrors > Home > ILE Home > Th. List > a16gb | GIF version |
Description: A generalization of Axiom ax-16 1801. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
a16gb | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | a16g 1851 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | |
2 | ax-4 1497 | . 2 ⊢ (∀𝑧𝜑 → 𝜑) | |
3 | 1, 2 | impbid1 141 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1340 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 |
This theorem is referenced by: (None) |
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