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Mirrors > Home > MPE Home > Th. List > reximddv2 | Structured version Visualization version GIF version |
Description: Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
Ref | Expression |
---|---|
reximddv2.1 | ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒) |
reximddv2.2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) |
Ref | Expression |
---|---|
reximddv2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reximddv2.1 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 415 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)) |
3 | 2 | reximdva 3276 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → ∃𝑦 ∈ 𝐵 𝜒)) |
4 | 3 | impr 457 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓)) → ∃𝑦 ∈ 𝐵 𝜒) |
5 | reximddv2.2 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) | |
6 | 4, 5 | reximddv 3277 | 1 ⊢ (𝜑 → ∃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: r19.29vva 3338 prmgaplem8 16396 cpmadugsumfi 21487 cpmidg2sum 21490 cayhamlem4 21498 ltgseg 26384 cgraswap 26608 cgracom 26610 cgratr 26611 flatcgra 26612 dfcgra2 26618 xrofsup 30494 prmunb2 40650 |
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