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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 2923 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑)) | |
2 | 1 | r2exlem 3041 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-ral 2901 df-rex 2902 |
This theorem is referenced by: reean 3085 elxp2 5056 rnoprab2 6642 elrnmpt2res 6672 oeeu 7570 omxpenlem 7946 axcnre 9864 hash2prb 13111 pmtrrn2 17703 fsumvma 24738 upgredg 25811 umgredg 25812 usgrarnedg 25913 spanuni 27787 5oalem7 27903 3oalem3 27907 elfuns 31192 ellines 31429 dalem20 33997 diblsmopel 35478 iunrelexpuztr 37030 |
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