Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > drnf2 | GIF version |
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Ref | Expression |
---|---|
drex2.1 | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
drnf2 | ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drex2.1 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | dral2 1694 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓)) |
3 | 1, 2 | imbi12d 233 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑧𝜑) ↔ (𝜓 → ∀𝑧𝜓))) |
4 | 3 | dral2 1694 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝜑 → ∀𝑧𝜑) ↔ ∀𝑧(𝜓 → ∀𝑧𝜓))) |
5 | df-nf 1422 | . 2 ⊢ (Ⅎ𝑧𝜑 ↔ ∀𝑧(𝜑 → ∀𝑧𝜑)) | |
6 | df-nf 1422 | . 2 ⊢ (Ⅎ𝑧𝜓 ↔ ∀𝑧(𝜓 → ∀𝑧𝜓)) | |
7 | 4, 5, 6 | 3bitr4g 222 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1314 Ⅎwnf 1421 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 |
This theorem depends on definitions: df-bi 116 df-nf 1422 |
This theorem is referenced by: nfsbxy 1895 nfsbxyt 1896 drnfc2 2276 |
Copyright terms: Public domain | W3C validator |