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Mirrors > Home > ILE Home > Th. List > sbralie | GIF version |
Description: Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.) |
Ref | Expression |
---|---|
sbralie.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbralie | ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvralsv 2602 | . . . 4 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) | |
2 | 1 | sbbii 1696 | . . 3 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑥 / 𝑦]∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) |
3 | nfv 1467 | . . . 4 ⊢ Ⅎ𝑦∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 | |
4 | raleq 2563 | . . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)) | |
5 | 3, 4 | sbie 1722 | . . 3 ⊢ ([𝑥 / 𝑦]∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑) |
6 | 2, 5 | bitri 183 | . 2 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑) |
7 | cbvralsv 2602 | . . 3 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) | |
8 | nfv 1467 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
9 | 8 | sbco2 1888 | . . . . 5 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
10 | nfv 1467 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
11 | sbralie.1 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
12 | 10, 11 | sbie 1722 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
13 | 9, 12 | bitri 183 | . . . 4 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ 𝜓) |
14 | 13 | ralbii 2385 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
15 | 7, 14 | bitri 183 | . 2 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
16 | 6, 15 | bitri 183 | 1 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 [wsb 1693 ∀wral 2360 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 |
This theorem is referenced by: (None) |
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