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Mirrors > Home > MPE Home > Th. List > r19.29vvaOLD | Structured version Visualization version GIF version |
Description: Obsolete version of r19.29vva 3336 as of 28-Jun-2023. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
r19.29vva.1 | ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒) |
r19.29vva.2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) |
Ref | Expression |
---|---|
r19.29vvaOLD | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.29vva.1 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 415 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)) |
3 | 2 | ralrimiva 3182 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 (𝜓 → 𝜒)) |
4 | 3 | ralrimiva 3182 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝜒)) |
5 | r19.29vva.2 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) | |
6 | 4, 5 | r19.29d2r 3335 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓)) |
7 | pm3.35 801 | . . . . 5 ⊢ ((𝜓 ∧ (𝜓 → 𝜒)) → 𝜒) | |
8 | 7 | ancoms 461 | . . . 4 ⊢ (((𝜓 → 𝜒) ∧ 𝜓) → 𝜒) |
9 | 8 | rexlimivw 3282 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓) → 𝜒) |
10 | 9 | rexlimivw 3282 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝜓 → 𝜒) ∧ 𝜓) → 𝜒) |
11 | 6, 10 | syl 17 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 |
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 3143 df-rex 3144 |
This theorem is referenced by: (None) |
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