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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 7955. (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 7955 | . 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝜑} ∈ V |
5 | 1, 4 | abrexex2 7955 | 1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 {cab 2703 ∃wrex 3064 Vcvv 3468 |
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 2697 ax-rep 5278 ax-sep 5292 ax-un 7722 |
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 2528 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-v 3470 df-in 3950 df-ss 3960 df-uni 4903 df-iun 4992 |
This theorem is referenced by: brdom7disj 10528 brdom6disj 10529 lineset 39122 |
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