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Mirrors > Home > ILE Home > Th. List > sbrbif | GIF version |
Description: Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
sbrbif.1 | ⊢ (𝜒 → ∀𝑥𝜒) |
sbrbif.2 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
sbrbif | ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbrbif.2 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | |
2 | 1 | sbrbis 1948 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜒) ↔ (𝜓 ↔ [𝑦 / 𝑥]𝜒)) |
3 | sbrbif.1 | . . . 4 ⊢ (𝜒 → ∀𝑥𝜒) | |
4 | 3 | sbh 1763 | . . 3 ⊢ ([𝑦 / 𝑥]𝜒 ↔ 𝜒) |
5 | 4 | bibi2i 226 | . 2 ⊢ ((𝜓 ↔ [𝑦 / 𝑥]𝜒) ↔ (𝜓 ↔ 𝜒)) |
6 | 2, 5 | bitri 183 | 1 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1340 [wsb 1749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 |
This theorem is referenced by: (None) |
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