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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcleqf | Structured version Visualization version GIF version |
Description: Equality connective between classes. Same as dfcleq 2604, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
dfcleqf.1 | ⊢ Ⅎ𝑥𝐴 |
dfcleqf.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
dfcleqf | ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2604 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) | |
2 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | dfcleqf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfel 2763 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
5 | dfcleqf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 2, 5 | nfel 2763 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
7 | 4, 6 | nfbi 1821 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) |
8 | nfv 1830 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) | |
9 | eleq1 2676 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
10 | eleq1 2676 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
11 | 9, 10 | bibi12d 334 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))) |
12 | 7, 8, 11 | cbval 2259 | . 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
13 | 1, 12 | bitri 263 | 1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∀wal 1473 = wceq 1475 ∈ 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: ssmapsn 38403 preimagelt 39589 preimalegt 39590 pimrecltpos 39596 pimrecltneg 39610 smfaddlem1 39649 |
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