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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.) Avoid ax-12 2175. (Revised by Wolf Lammen, 19-Jun-2023.) |
Ref | Expression |
---|---|
nfceqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
nfceqi | ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfceqi.1 | . . . . 5 ⊢ 𝐴 = 𝐵 | |
2 | 1 | eleq2i 2831 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) |
3 | 2 | nfbii 1849 | . . 3 ⊢ (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵) |
4 | 3 | albii 1816 | . 2 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵) |
5 | df-nfc 2890 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
6 | df-nfc 2890 | . 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵) | |
7 | 4, 5, 6 | 3bitr4i 303 | 1 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∀wal 1535 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 df-cleq 2727 df-clel 2814 df-nfc 2890 |
This theorem is referenced by: nfcxfr 2901 nfcxfrd 2902 |
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