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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 3194 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑)) | |
2 | 1 | r2exlem 3143 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 ∃wex 1781 ∈ wcel 2106 ∃wrex 3070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-ral 3062 df-rex 3071 |
This theorem is referenced by: reeanlem 3225 elxp2 5699 elinxp 6017 rnoprab2 7509 elrnmpores 7542 oeeu 8599 omxpenlem 9069 axcnre 11155 hash2prb 14429 hashle2prv 14435 pmtrrn2 19322 fsumvma 26705 umgredg 28387 fusgr2wsp2nb 29576 spanuni 30784 5oalem7 30900 3oalem3 30904 trsp2cyc 32269 fmla0xp 34362 elfuns 34875 ellines 35112 dalem20 38552 diblsmopel 40030 iunrelexpuztr 42455 sprssspr 46135 prprelb 46170 |
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