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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 1651 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
3 | 2 | equcoms 1594 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 → [𝑦 / 𝑥]𝜑)) |
4 | 3 | com12 27 | . . 3 ⊢ (𝜑 → (𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
5 | 1, 4 | alrimih 1358 | . 2 ⊢ (𝜑 → ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
6 | sb2 1650 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑) | |
7 | 1 | sbid2h 1729 | . . 3 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑) |
8 | 6, 7 | sylib 127 | . 2 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) → 𝜑) |
9 | 5, 8 | impbii 117 | 1 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 [wsb 1645 |
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-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-sb 1646 |
This theorem is referenced by: 2sb6rf 1866 eu1 1925 |
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