Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > drnfc2 | GIF version |
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.) |
Ref | Expression |
---|---|
drnfc1.1 | ⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
drnfc2 | ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drnfc1.1 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) | |
2 | 1 | eleq2d 2107 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵)) |
3 | 2 | drnf2 1622 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧 𝑤 ∈ 𝐴 ↔ Ⅎ𝑧 𝑤 ∈ 𝐵)) |
4 | 3 | dral2 1619 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐴 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐵)) |
5 | df-nfc 2167 | . 2 ⊢ (Ⅎ𝑧𝐴 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐴) | |
6 | df-nfc 2167 | . 2 ⊢ (Ⅎ𝑧𝐵 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐵) | |
7 | 4, 5, 6 | 3bitr4g 212 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 Ⅎwnfc 2165 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-cleq 2033 df-clel 2036 df-nfc 2167 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |