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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 2148 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵)) |
3 | 2 | drnf2 1662 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧 𝑤 ∈ 𝐴 ↔ Ⅎ𝑧 𝑤 ∈ 𝐵)) |
4 | 3 | dral2 1659 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐴 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐵)) |
5 | df-nfc 2208 | . 2 ⊢ (Ⅎ𝑧𝐴 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐴) | |
6 | df-nfc 2208 | . 2 ⊢ (Ⅎ𝑧𝐵 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐵) | |
7 | 4, 5, 6 | 3bitr4g 221 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 = wceq 1284 Ⅎwnf 1389 ∈ wcel 1433 Ⅎwnfc 2206 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-cleq 2074 df-clel 2077 df-nfc 2208 |
This theorem is referenced by: (None) |
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