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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 3188 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ 𝜑)) | |
2 | 1 | r2exlem 3137 | 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 397 ∃wex 1782 ∈ wcel 2107 ∃wrex 3070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-ral 3062 df-rex 3071 |
This theorem is referenced by: reeanlem 3215 elxp2 5658 elinxp 5976 rnoprab2 7462 elrnmpores 7494 oeeu 8551 omxpenlem 9020 axcnre 11105 hash2prb 14377 hashle2prv 14383 pmtrrn2 19247 fsumvma 26577 umgredg 28131 fusgr2wsp2nb 29320 spanuni 30528 5oalem7 30644 3oalem3 30648 trsp2cyc 32021 fmla0xp 34034 elfuns 34546 ellines 34783 dalem20 38202 diblsmopel 39680 iunrelexpuztr 42079 sprssspr 45759 prprelb 45794 |
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