Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > hbsb4 | GIF version |
Description: A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.) |
Ref | Expression |
---|---|
hbsb4.1 | ⊢ (𝜑 → ∀𝑧𝜑) |
Ref | Expression |
---|---|
hbsb4 | ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbsb4.1 | . . 3 ⊢ (𝜑 → ∀𝑧𝜑) | |
2 | 1 | hbsb 1936 | . 2 ⊢ ([𝑤 / 𝑥]𝜑 → ∀𝑧[𝑤 / 𝑥]𝜑) |
3 | sbequ 1827 | . 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 2, 3 | dvelimALT 1997 | 1 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀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-in2 605 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-bndl 1496 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: hbsb4t 2000 dvelimf 2002 |
Copyright terms: Public domain | W3C validator |