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Mirrors > Home > ILE Home > Th. List > r19.40 | GIF version |
Description: Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.) |
Ref | Expression |
---|---|
r19.40 | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | reximi 2458 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐴 𝜑) |
3 | simpr 108 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓) | |
4 | 3 | reximi 2458 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → ∃𝑥 ∈ 𝐴 𝜓) |
5 | 2, 4 | jca 300 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∃wrex 2349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-ral 2353 df-rex 2354 |
This theorem is referenced by: rexanuz 9874 |
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