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Mirrors > Home > ILE Home > Th. List > sb5rf | GIF version |
Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
sb5rf.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
Ref | Expression |
---|---|
sb5rf | ⊢ (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb5rf.1 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | sbid2h 1729 | . . 3 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑) |
3 | sb1 1649 | . . 3 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 → ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑)) | |
4 | 2, 3 | sylbir 125 | . 2 ⊢ (𝜑 → ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑)) |
5 | stdpc7 1653 | . . . 4 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 → 𝜑)) | |
6 | 5 | imp 115 | . . 3 ⊢ ((𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) → 𝜑) |
7 | 1, 6 | exlimih 1484 | . 2 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑) → 𝜑) |
8 | 4, 7 | impbii 117 | 1 ⊢ (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 ∃wex 1381 [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: 2sb5rf 1865 sbelx 1873 |
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