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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 1770). New proofs should use sbied 1768 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 1746 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) | |
2 | sbiedh.1 | . . . . 5 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | sbiedh.3 | . . . . . . 7 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) | |
4 | biimp 117 | . . . . . . 7 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒)) | |
5 | 3, 4 | syl6 33 | . . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) |
6 | 5 | impd 252 | . . . . 5 ⊢ (𝜑 → ((𝑥 = 𝑦 ∧ 𝜓) → 𝜒)) |
7 | 2, 6 | eximdh 1591 | . . . 4 ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑥𝜒)) |
8 | 1, 7 | syl5 32 | . . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∃𝑥𝜒)) |
9 | sbiedh.2 | . . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
10 | 2, 9 | 19.9hd 1642 | . . 3 ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒)) |
11 | 8, 10 | syld 45 | . 2 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒)) |
12 | biimpr 129 | . . . . . . 7 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓)) | |
13 | 3, 12 | syl6 33 | . . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜒 → 𝜓))) |
14 | 13 | com23 78 | . . . . 5 ⊢ (𝜑 → (𝜒 → (𝑥 = 𝑦 → 𝜓))) |
15 | 2, 14 | alimdh 1447 | . . . 4 ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑥(𝑥 = 𝑦 → 𝜓))) |
16 | sb2 1747 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) → [𝑦 / 𝑥]𝜓) | |
17 | 15, 16 | syl6 33 | . . 3 ⊢ (𝜑 → (∀𝑥𝜒 → [𝑦 / 𝑥]𝜓)) |
18 | 9, 17 | syld 45 | . 2 ⊢ (𝜑 → (𝜒 → [𝑦 / 𝑥]𝜓)) |
19 | 11, 18 | impbid 128 | 1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1333 ∃wex 1472 [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-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-sb 1743 |
This theorem is referenced by: sbied 1768 sbieh 1770 sbcomxyyz 1952 |
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