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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 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvralsv 2668 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓) | |
2 | 1 | sbbii 1738 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓) |
3 | nfv 1508 | . . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 | |
4 | raleq 2626 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓)) | |
5 | 3, 4 | sbie 1764 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓) |
6 | cbvralsv 2668 | . . 3 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓) | |
7 | nfv 1508 | . . . . . 6 ⊢ Ⅎ𝑧𝜓 | |
8 | 7 | sbco2 1938 | . . . . 5 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ [𝑥 / 𝑦]𝜓) |
9 | nfv 1508 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
10 | sbralie.1 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
11 | 10 | bicomd 140 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑)) |
12 | 11 | equcoms 1684 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑)) |
13 | 9, 12 | sbie 1764 | . . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑) |
14 | 8, 13 | bitri 183 | . . . 4 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ 𝜑) |
15 | 14 | ralbii 2441 | . . 3 ⊢ (∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 𝜑) |
16 | 6, 15 | bitri 183 | . 2 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 𝜑) |
17 | 2, 5, 16 | 3bitrri 206 | 1 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 [wsb 1735 ∀wral 2416 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 |
This theorem is referenced by: (None) |
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