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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 as in dfcleq 2815. See also cleqh 2936. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) Avoid ax-13 2386. (Revised by Wolf Lammen, 10-May-2023.) Avoid ax-10 2141. (Revised by Gino Giotto, 20-Aug-2023.) |
Ref | Expression |
---|---|
cleqf.1 | ⊢ Ⅎ𝑥𝐴 |
cleqf.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
cleqf | ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2815 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) | |
2 | nfv 1911 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) | |
3 | cleqf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfcriv 2967 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
5 | cleqf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | nfcriv 2967 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
7 | 4, 6 | nfbi 1900 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) |
8 | eleq1w 2895 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
9 | eleq1w 2895 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) | |
10 | 8, 9 | bibi12d 348 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))) |
11 | 2, 7, 10 | cbvalv1 2357 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
12 | 1, 11 | bitr4i 280 | 1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1531 = wceq 1533 ∈ wcel 2110 Ⅎwnfc 2961 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-cleq 2814 df-clel 2893 df-nfc 2963 |
This theorem is referenced by: abid2f 3012 abeq2f 3013 eqvf 3503 eqrd 3985 eq0f 4304 iunab 4967 iinab 4982 mbfposr 24247 mbfinf 24260 itg1climres 24309 bnj1366 32096 bj-rabtrALT 34244 bj-rcleqf 34332 compab 40767 ssmapsn 41472 infnsuprnmpt 41515 pimrecltpos 42981 pimrecltneg 42995 smfaddlem1 43033 smflimsuplem7 43094 absnsb 43256 |
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