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Mirrors > Home > MPE Home > Th. List > 2r19.29 | Structured version Visualization version GIF version |
Description: Theorem r19.29 3210 with two quantifiers. (Contributed by Rodolfo Medina, 25-Sep-2010.) |
Ref | Expression |
---|---|
2r19.29 | ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.29 3210 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) | |
2 | r19.29 3210 | . . 3 ⊢ ((∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓)) | |
3 | 2 | reximi 3149 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓)) |
4 | 1, 3 | syl 17 | 1 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wral 3050 ∃wrex 3051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 |
This theorem depends on definitions: df-bi 197 df-an 385 df-ex 1854 df-ral 3055 df-rex 3056 |
This theorem is referenced by: prter2 34688 |
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