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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. See also cleqh 2873. (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 2906 | . 2 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
3 | cleqf.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 3 | nfcrii 2906 | . 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) |
5 | 2, 4 | cleqh 2873 | 1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∀wal 1629 = wceq 1631 ∈ wcel 2145 Ⅎwnfc 2900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-cleq 2764 df-clel 2767 df-nfc 2902 |
This theorem is referenced by: abid2f 2940 eqvf 3355 eqrd 3771 eq0f 4073 iunab 4700 iinab 4715 mbfposr 23639 mbfinf 23652 itg1climres 23701 bnj1366 31238 bj-rabtrALT 33259 compab 39171 dfcleqf 39776 |
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