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Mirrors > Home > MPE Home > Th. List > rexrab | Structured version Visualization version GIF version |
Description: Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
ralab.1 | ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexrab | ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralab.1 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) | |
2 | 1 | elrab 3331 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)) |
3 | 2 | anbi1i 727 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∧ 𝜒)) |
4 | anass 679 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜓) ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜒))) | |
5 | 3, 4 | bitri 263 | . 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜒))) |
6 | 5 | rexbii2 3021 | 1 ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∃wrex 2897 {crab 2900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 |
This theorem is referenced by: wereu2 5035 wdom2d 8368 enfin2i 9026 infm3 10861 pmtrfrn 17701 pgpssslw 17852 ellspd 19960 1stcfb 21058 xkobval 21199 xkococn 21273 imasdsf1olem 21988 nbgraf1olem1 25970 rusgranumwlks 26483 cvmliftlem15 30534 wsuclem 31017 wsuclemOLD 31018 poimirlem4 32583 poimirlem26 32605 poimirlem27 32606 rexrabdioph 36376 hbtlem6 36718 rusgrnumwwlks 41177 |
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