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Mirrors > Home > ILE Home > Th. List > sb6rf | GIF version |
Description: Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
sb5rf.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
Ref | Expression |
---|---|
sb6rf | ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb5rf.1 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | sbequ1 1768 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
3 | 2 | equcoms 1708 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 → [𝑦 / 𝑥]𝜑)) |
4 | 3 | com12 30 | . . 3 ⊢ (𝜑 → (𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
5 | 1, 4 | alrimih 1469 | . 2 ⊢ (𝜑 → ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
6 | sb2 1767 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑) | |
7 | 1 | sbid2h 1849 | . . 3 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑) |
8 | 6, 7 | sylib 122 | . 2 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → 𝜑) |
9 | 5, 8 | impbii 126 | 1 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 [wsb 1762 |
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-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-sb 1763 |
This theorem is referenced by: 2sb6rf 1990 eu1 2051 |
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