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Mirrors > Home > MPE Home > Th. List > Mathboxes > rexlimd3 | Structured version Visualization version GIF version |
Description: * Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
rexlimd3.1 | ⊢ Ⅎ𝑥𝜑 |
rexlimd3.2 | ⊢ Ⅎ𝑥𝜒 |
rexlimd3.3 | ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
rexlimd3 | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimd3.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | rexlimd3.2 | . 2 ⊢ Ⅎ𝑥𝜒 | |
3 | rexlimd3.3 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) | |
4 | 3 | exp31 420 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
5 | 1, 2, 4 | rexlimd 3314 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 Ⅎwnf 1775 ∈ wcel 2105 ∃wrex 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-ral 3140 df-rex 3141 |
This theorem is referenced by: dffo3f 41314 |
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