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Mirrors > Home > MPE Home > Th. List > nfceqi | Structured version Visualization version GIF version |
Description: Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
Ref | Expression |
---|---|
nfceqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
nfceqi | ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1721 | . . 3 ⊢ Ⅎ𝑥⊤ | |
2 | nfceqi.1 | . . . 4 ⊢ 𝐴 = 𝐵 | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → 𝐴 = 𝐵) |
4 | 1, 3 | nfceqdf 2747 | . 2 ⊢ (⊤ → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) |
5 | 4 | trud 1484 | 1 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ⊤wtru 1476 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-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 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: nfcxfr 2749 nfcxfrd 2750 |
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