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Mirrors > Home > MPE Home > Th. List > reximdvva | Structured version Visualization version GIF version |
Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by AV, 5-Jan-2022.) |
Ref | Expression |
---|---|
reximdvva.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
reximdvva | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reximdvva.1 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒)) | |
2 | 1 | anassrs 470 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)) |
3 | 2 | reximdva 3276 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → ∃𝑦 ∈ 𝐵 𝜒)) |
4 | 3 | reximdva 3276 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∃wrex 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-ral 3145 df-rex 3146 |
This theorem is referenced by: reuop 6146 lcmgcdlem 15952 lsmelval2 19859 cpmadugsum 21488 axpasch 26729 frgrwopreglem5 28102 frgrwopreglem5ALT 28103 eulerpartlemgvv 31636 cusgr3cyclex 32385 cvmlift2lem10 32561 ftc1anclem6 34974 prprelprb 43686 reupr 43691 |
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