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Mirrors > Home > MPE Home > Th. List > rabeq2i | Structured version Visualization version GIF version |
Description: Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.) |
Ref | Expression |
---|---|
rabeq2i.1 | ⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Ref | Expression |
---|---|
rabeq2i | ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeq2i.1 | . . 3 ⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑} | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑}) |
3 | rabid 3379 | . 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) | |
4 | 2, 3 | bitri 277 | 1 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1536 df-ex 1777 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-rab 3147 |
This theorem is referenced by: fvmptss 6775 tfis 7563 nqereu 10345 rpnnen1lem2 12370 rpnnen1lem1 12371 rpnnen1lem3 12372 rpnnen1lem5 12374 qustgpopn 22722 nbusgrf1o0 27145 finsumvtxdg2ssteplem3 27323 frgrwopreglem2 28086 frgrwopreglem5lem 28093 resf1o 30460 ballotlem2 31741 reprsuc 31881 oddprm2 31921 hgt750lemb 31922 bnj1476 32114 bnj1533 32119 bnj1538 32122 bnj1523 32338 cvmlift2lem12 32556 neibastop2lem 33703 topdifinfindis 34621 topdifinffinlem 34622 stoweidlem24 42302 stoweidlem31 42309 stoweidlem52 42330 stoweidlem54 42332 stoweidlem57 42335 salexct 42610 ovolval5lem3 42929 pimdecfgtioc 42986 pimincfltioc 42987 pimdecfgtioo 42988 pimincfltioo 42989 smfsuplem1 43078 smfsuplem3 43080 smfliminflem 43097 prprsprreu 43674 |
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