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Mirrors > Home > ILE Home > Th. List > sbiedh | GIF version |
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1673). New proofs should use sbied 1671 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbiedh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
sbiedh.2 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
sbiedh.3 | ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
sbiedh | ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb1 1649 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) | |
2 | sbiedh.1 | . . . . 5 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | sbiedh.3 | . . . . . . 7 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
4 | bi1 111 | . . . . . . 7 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒)) | |
5 | 3, 4 | syl6 29 | . . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) |
6 | 5 | impd 242 | . . . . 5 ⊢ (𝜑 → ((𝑥 = 𝑦 ∧ 𝜓) → 𝜒)) |
7 | 2, 6 | eximdh 1502 | . . . 4 ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑥𝜒)) |
8 | 1, 7 | syl5 28 | . . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∃𝑥𝜒)) |
9 | sbiedh.2 | . . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
10 | 2, 9 | 19.9hd 1552 | . . 3 ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒)) |
11 | 8, 10 | syld 40 | . 2 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒)) |
12 | bi2 121 | . . . . . . 7 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓)) | |
13 | 3, 12 | syl6 29 | . . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜒 → 𝜓))) |
14 | 13 | com23 72 | . . . . 5 ⊢ (𝜑 → (𝜒 → (𝑥 = 𝑦 → 𝜓))) |
15 | 2, 14 | alimdh 1356 | . . . 4 ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑥(𝑥 = 𝑦 → 𝜓))) |
16 | sb2 1650 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) → [𝑦 / 𝑥]𝜓) | |
17 | 15, 16 | syl6 29 | . . 3 ⊢ (𝜑 → (∀𝑥𝜒 → [𝑦 / 𝑥]𝜓)) |
18 | 9, 17 | syld 40 | . 2 ⊢ (𝜑 → (𝜒 → [𝑦 / 𝑥]𝜓)) |
19 | 11, 18 | impbid 120 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 ∃wex 1381 [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-sb 1646 |
This theorem is referenced by: sbied 1671 sbieh 1673 sbcomxyyz 1846 |
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