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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 7844. (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 7844 | . 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝜑} ∈ V |
5 | 1, 4 | abrexex2 7844 | 1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2104 {cab 2713 ∃wrex 3071 Vcvv 3437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-tru 1542 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-v 3439 df-in 3899 df-ss 3909 df-uni 4845 df-iun 4933 |
This theorem is referenced by: brdom7disj 10333 brdom6disj 10334 lineset 37794 |
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