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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 2813. See also cleqh 2934. (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) Avoid ax-13 2384. (Revised by Wolf Lammen, 10-May-2023.) Avoid ax-10 2139. (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 2813 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) | |
2 | nfv 1909 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) | |
3 | cleqf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfcriv 2965 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
5 | cleqf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | nfcriv 2965 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
7 | 4, 6 | nfbi 1898 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) |
8 | eleq1w 2893 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
9 | eleq1w 2893 | . . . 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 2355 | . 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 1529 = wceq 1531 ∈ wcel 2108 Ⅎwnfc 2959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-cleq 2812 df-clel 2891 df-nfc 2961 |
This theorem is referenced by: abid2f 3010 abeq2f 3011 eqvf 3502 eqrd 3984 eq0f 4303 iunab 4966 iinab 4981 mbfposr 24245 mbfinf 24258 itg1climres 24307 bnj1366 32094 bj-rabtrALT 34242 bj-rcleqf 34330 compab 40765 ssmapsn 41469 infnsuprnmpt 41512 pimrecltpos 42978 pimrecltneg 42992 smfaddlem1 43030 smflimsuplem7 43091 absnsb 43253 |
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