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Mirrors > Home > MPE Home > Th. List > r2ex | Structured version Visualization version GIF version |
Description: Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 10-Jan-2020.) |
Ref | Expression |
---|---|
r2ex | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r2al 3201 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑)) | |
2 | 1 | r2exlem 3302 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∃wex 1780 ∈ wcel 2114 ∃wrex 3139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-ral 3143 df-rex 3144 |
This theorem is referenced by: rexcomOLD 3356 reeanlem 3365 elxp2 5579 elinxp 5890 rnoprab2 7258 elrnmpores 7288 oeeu 8229 omxpenlem 8618 axcnre 10586 hash2prb 13831 hashle2prv 13837 pmtrrn2 18588 fsumvma 25789 umgredg 26923 fusgr2wsp2nb 28113 spanuni 29321 5oalem7 29437 3oalem3 29441 trsp2cyc 30765 fmla0xp 32630 elfuns 33376 ellines 33613 dalem20 36844 diblsmopel 38322 iunrelexpuztr 40084 sprssspr 43663 prprelb 43698 |
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