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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 449 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)) |
3 | 2 | reximdva 3000 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → ∃𝑦 ∈ 𝐵 𝜒)) |
4 | 3 | impr 647 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜓)) → ∃𝑦 ∈ 𝐵 𝜒) |
5 | reximddv2.2 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) | |
6 | 4, 5 | reximddv 3001 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-ral 2901 df-rex 2902 |
This theorem is referenced by: prmgaplem8 15600 cpmadugsumfi 20501 cpmidg2sum 20504 cayhamlem4 20512 ltgseg 25291 cgraswap 25512 cgracom 25514 cgratr 25515 dfcgra2 25521 xrofsup 28923 prmunb2 37532 |
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