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Mirrors > Home > ILE Home > Th. List > sbied | GIF version |
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1674). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Ref | Expression |
---|---|
sbied.1 | ⊢ Ⅎ𝑥𝜑 |
sbied.2 | ⊢ (𝜑 → Ⅎ𝑥𝜒) |
sbied.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
sbied | ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbied.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfri 1412 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | sbied.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
4 | 3 | nfrd 1413 | . 2 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
5 | sbied.3 | . 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
6 | 2, 4, 5 | sbiedh 1670 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 Ⅎwnf 1349 [wsb 1645 |
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 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 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: sbiedv 1672 dvelimdf 1892 cbvrald 9927 |
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