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Mirrors > Home > ILE Home > Th. List > sbalyz | GIF version |
Description: Move universal quantifier in and out of substitution. Identical to sbal 2012 except that it has an additional distinct variable constraint on 𝑦 and 𝑧. (Contributed by Jim Kingdon, 29-Dec-2017.) |
Ref | Expression |
---|---|
sbalyz | ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 1552 | . . . 4 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
2 | 1 | nfsbxy 1954 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]∀𝑥𝜑 |
3 | ax-4 1521 | . . . 4 ⊢ (∀𝑥𝜑 → 𝜑) | |
4 | 3 | sbimi 1775 | . . 3 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 → [𝑧 / 𝑦]𝜑) |
5 | 2, 4 | alrimi 1533 | . 2 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑) |
6 | sb6 1898 | . . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜑)) | |
7 | 6 | albii 1481 | . . . 4 ⊢ (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦(𝑦 = 𝑧 → 𝜑)) |
8 | alcom 1489 | . . . 4 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑧 → 𝜑) ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑)) | |
9 | 7, 8 | bitri 184 | . . 3 ⊢ (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑)) |
10 | nfv 1539 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
11 | 10 | stdpc5 1595 | . . . . 5 ⊢ (∀𝑥(𝑦 = 𝑧 → 𝜑) → (𝑦 = 𝑧 → ∀𝑥𝜑)) |
12 | 11 | alimi 1466 | . . . 4 ⊢ (∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑) → ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)) |
13 | sb2 1778 | . . . 4 ⊢ (∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑) → [𝑧 / 𝑦]∀𝑥𝜑) | |
14 | 12, 13 | syl 14 | . . 3 ⊢ (∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑) → [𝑧 / 𝑦]∀𝑥𝜑) |
15 | 9, 14 | sylbi 121 | . 2 ⊢ (∀𝑥[𝑧 / 𝑦]𝜑 → [𝑧 / 𝑦]∀𝑥𝜑) |
16 | 5, 15 | impbii 126 | 1 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1362 [wsb 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 |
This theorem is referenced by: sbal 2012 |
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