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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 2544 | . . . 4 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) | |
2 | 1 | sbbii 1648 | . . 3 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑥 / 𝑦]∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑) |
3 | nfv 1421 | . . . 4 ⊢ Ⅎ𝑦∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 | |
4 | raleq 2505 | . . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)) | |
5 | 3, 4 | sbie 1674 | . . 3 ⊢ ([𝑥 / 𝑦]∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑) |
6 | 2, 5 | bitri 173 | . 2 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑) |
7 | cbvralsv 2544 | . . 3 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) | |
8 | nfv 1421 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
9 | 8 | sbco2 1839 | . . . . 5 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
10 | nfv 1421 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
11 | sbralie.1 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
12 | 10, 11 | sbie 1674 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
13 | 9, 12 | bitri 173 | . . . 4 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ 𝜓) |
14 | 13 | ralbii 2330 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
15 | 7, 14 | bitri 173 | . 2 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
16 | 6, 15 | bitri 173 | 1 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 [wsb 1645 ∀wral 2306 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 |
This theorem is referenced by: (None) |
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