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Mirrors > Home > MPE Home > Th. List > Mathboxes > reximdd | Structured version Visualization version GIF version |
Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
reximdd.1 | ⊢ Ⅎ𝑥𝜑 |
reximdd.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) |
reximdd.3 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
Ref | Expression |
---|---|
reximdd | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reximdd.3 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | |
2 | reximdd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | reximdd.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) | |
4 | 3 | 3exp 1120 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
5 | 2, 4 | reximdai 3221 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) |
6 | 1, 5 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 Ⅎwnf 1790 ∈ wcel 2114 ∃wrex 3054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1090 df-ex 1787 df-nf 1791 df-ral 3058 df-rex 3059 |
This theorem is referenced by: xlimmnfvlem2 42916 xlimmnfv 42917 xlimpnfvlem2 42920 xlimpnfv 42921 |
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