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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 2211 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
4 | eqrd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | eqrd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | cleqf 2932 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
7 | 3, 6 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-cleq 2727 df-clel 2814 df-nfc 2890 |
This theorem is referenced by: eqri 4016 eqrrabd 4096 sniota 6554 fimarab 6983 dissnlocfin 23553 imasnopn 23714 imasncld 23715 imasncls 23716 blval2 24591 ofpreima 32682 algextdeglem6 33728 constrfin 33751 zarcls 33835 ordtconnlem1 33885 qqhval2 33945 reprdifc 34621 topdifinfindis 37329 icorempo 37334 isbasisrelowllem1 37338 isbasisrelowllem2 37339 sticksstones11 42138 areaquad 43205 rfcnpre1 44957 rfcnpre2 44969 preimagelt 46655 preimalegt 46656 |
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