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Mirrors > Home > MPE Home > Th. List > cleqh | 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 cleqf 3010. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.) Remove dependency on ax-13 2390. (Revised by BJ, 30-Nov-2020.) |
Ref | Expression |
---|---|
cleqh.1 | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
cleqh.2 | ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) |
Ref | Expression |
---|---|
cleqh | ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcleq 2815 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) | |
2 | nfv 1915 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) | |
3 | cleqh.1 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
4 | 3 | nf5i 2150 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
5 | cleqh.2 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) | |
6 | 5 | nf5i 2150 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
7 | 4, 6 | nfbi 1904 | . . 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 2361 | . 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: → wi 4 ↔ wb 208 ∀wal 1535 = wceq 1537 ∈ wcel 2114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-cleq 2814 df-clel 2893 |
This theorem is referenced by: abeq2 2945 abbiOLD 2955 cleqfOLD 3011 abeq2fOLD 3014 |
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