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Mirrors > Home > MPE Home > Th. List > eqrd | Structured version Visualization version GIF version |
Description: Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.) (Proof shortened by BJ, 1-Dec-2021.) |
Ref | Expression |
---|---|
eqrd.0 | ⊢ Ⅎ𝑥𝜑 |
eqrd.1 | ⊢ Ⅎ𝑥𝐴 |
eqrd.2 | ⊢ Ⅎ𝑥𝐵 |
eqrd.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
Ref | Expression |
---|---|
eqrd | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrd.0 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | eqrd.3 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | alrimi 2212 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
4 | eqrd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | eqrd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | cleqf 3013 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
7 | 3, 6 | sylibr 236 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1534 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 Ⅎwnfc 2964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-cleq 2817 df-clel 2896 df-nfc 2966 |
This theorem is referenced by: eqri 3990 sniota 6349 dissnlocfin 22140 imasnopn 22301 imasncld 22302 imasncls 22303 blval2 23175 fimarab 30393 ofpreima 30413 ordtconnlem1 31171 qqhval2 31227 reprdifc 31902 topdifinfindis 34631 icorempo 34636 isbasisrelowllem1 34640 isbasisrelowllem2 34641 areaquad 39829 rfcnpre1 41282 rfcnpre2 41294 preimagelt 42987 preimalegt 42988 |
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