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Mirrors > Home > MPE Home > Th. List > ab2rexex2 | Structured version Visualization version GIF version |
Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7950. (Contributed by NM, 20-Sep-2011.) |
Ref | Expression |
---|---|
ab2rexex2.1 | ⊢ 𝐴 ∈ V |
ab2rexex2.2 | ⊢ 𝐵 ∈ V |
ab2rexex2.3 | ⊢ {𝑧 ∣ 𝜑} ∈ V |
Ref | Expression |
---|---|
ab2rexex2 | ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ab2rexex2.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | ab2rexex2.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | ab2rexex2.3 | . . 3 ⊢ {𝑧 ∣ 𝜑} ∈ V | |
4 | 2, 3 | abrexex2 7950 | . 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝜑} ∈ V |
5 | 1, 4 | abrexex2 7950 | 1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 {cab 2701 ∃wrex 3062 Vcvv 3466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-v 3468 df-in 3948 df-ss 3958 df-uni 4901 df-iun 4990 |
This theorem is referenced by: brdom7disj 10523 brdom6disj 10524 lineset 39103 |
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