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Mirrors > Home > MPE Home > Th. List > drnfc1 | Structured version Visualization version 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 |
---|---|
drnfc1 | ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drnfc1.1 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵) | |
2 | 1 | eleq2d 2673 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵)) |
3 | 2 | drnf1 2317 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ Ⅎ𝑦 𝑤 ∈ 𝐵)) |
4 | 3 | dral2 2312 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵)) |
5 | df-nfc 2740 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴) | |
6 | df-nfc 2740 | . 2 ⊢ (Ⅎ𝑦𝐵 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵) | |
7 | 4, 5, 6 | 3bitr4g 302 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-cleq 2603 df-clel 2606 df-nfc 2740 |
This theorem is referenced by: nfabd2 2770 nfcvb 4824 nfriotad 6519 bj-nfcsym 32079 |
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