Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cleqf | Structured version Visualization version GIF version |
Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2711. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) |
Ref | Expression |
---|---|
cleqf.1 | ⊢ Ⅎ𝑥𝐴 |
cleqf.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
cleqf | ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cleqf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcrii 2744 | . 2 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
3 | cleqf.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 3 | nfcrii 2744 | . 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) |
5 | 2, 4 | cleqh 2711 | 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-sb 1868 df-cleq 2603 df-clel 2606 df-nfc 2740 |
This theorem is referenced by: abid2f 2777 eqvf 3177 eq0f 3884 n0fOLD 3887 iunab 4502 iinab 4517 mbfposr 23225 mbfinf 23238 itg1climres 23287 bnj1366 30154 bj-rabtrALT 32119 compab 37666 |
Copyright terms: Public domain | W3C validator |