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Mirrors > Home > ILE Home > Th. List > drnf1 | 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 |
---|---|
drnf1 | ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drex2.1 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | dral1 1710 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
3 | 1, 2 | imbi12d 233 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑦𝜓))) |
4 | 3 | dral1 1710 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜓 → ∀𝑦𝜓))) |
5 | df-nf 1441 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
6 | df-nf 1441 | . 2 ⊢ (Ⅎ𝑦𝜓 ↔ ∀𝑦(𝜓 → ∀𝑦𝜓)) | |
7 | 4, 5, 6 | 3bitr4g 222 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1333 Ⅎwnf 1440 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1441 |
This theorem is referenced by: drnfc1 2316 |
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