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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 2202 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
4 | eqrd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | eqrd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | cleqf 2924 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
7 | 3, 6 | sylibr 233 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 Ⅎwnfc 2876 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-nf 1779 df-cleq 2718 df-clel 2803 df-nfc 2878 |
This theorem is referenced by: eqri 4000 sniota 6545 dissnlocfin 23524 imasnopn 23685 imasncld 23686 imasncls 23687 blval2 24562 eqrrabd 32429 fimarab 32560 ofpreima 32582 algextdeglem6 33589 zarcls 33689 ordtconnlem1 33739 qqhval2 33797 reprdifc 34473 topdifinfindis 37053 icorempo 37058 isbasisrelowllem1 37062 isbasisrelowllem2 37063 sticksstones11 41854 areaquad 42881 rfcnpre1 44618 rfcnpre2 44630 preimagelt 46320 preimalegt 46321 |
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