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Mirrors > Home > ILE Home > Th. List > sbhb | GIF version |
Description: Two ways of expressing "𝑥 is (effectively) not free in 𝜑." (Contributed by NM, 29-May-2009.) |
Ref | Expression |
---|---|
sbhb | ⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1506 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | sb8h 1834 | . . 3 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
3 | 2 | imbi2i 225 | . 2 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)) |
4 | 19.21v 1853 | . 2 ⊢ (∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)) | |
5 | 3, 4 | bitr4i 186 | 1 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1333 [wsb 1742 |
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-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-11 1486 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 |
This theorem is referenced by: sbnf2 1961 |
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