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Mirrors > Home > ILE Home > Th. List > r19.29d2r | GIF version |
Description: Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
Ref | Expression |
---|---|
r19.29d2r.1 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
r19.29d2r.2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) |
Ref | Expression |
---|---|
r19.29d2r | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.29d2r.1 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) | |
2 | r19.29d2r.2 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) | |
3 | r19.29 2594 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒)) | |
4 | 1, 2, 3 | syl2anc 409 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒)) |
5 | r19.29 2594 | . . 3 ⊢ ((∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒)) | |
6 | 5 | reximi 2554 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒)) |
7 | 4, 6 | syl 14 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wral 2435 ∃wrex 2436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-ral 2440 df-rex 2441 |
This theorem is referenced by: r19.29vva 2602 cauappcvgprlemdisj 7573 |
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