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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 2935 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
7 | 3, 6 | sylibr 237 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-cleq 2729 df-clel 2816 df-nfc 2886 |
This theorem is referenced by: eqri 3918 sniota 6368 dissnlocfin 22423 imasnopn 22584 imasncld 22585 imasncls 22586 blval2 23457 eqrrabd 30565 fimarab 30696 ofpreima 30719 zarcls 31535 ordtconnlem1 31585 qqhval2 31641 reprdifc 32316 topdifinfindis 35251 icorempo 35256 isbasisrelowllem1 35260 isbasisrelowllem2 35261 sticksstones11 39832 areaquad 40748 rfcnpre1 42233 rfcnpre2 42245 preimagelt 43909 preimalegt 43910 |
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