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Mirrors > Home > MPE Home > Th. List > r19.12OLD | Structured version Visualization version GIF version |
Description: Obsolete version of r19.12 3323 as of 19-May-2023. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
r19.12OLD | ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2976 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
2 | nfra1 3218 | . . . 4 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑 | |
3 | 1, 2 | nfrex 3308 | . . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
4 | ax-1 6 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)) | |
5 | 3, 4 | ralrimi 3215 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) |
6 | rsp 3204 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑)) | |
7 | 6 | com12 32 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 𝜑 → 𝜑)) |
8 | 7 | reximdv 3272 | . . 3 ⊢ (𝑦 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
9 | 8 | ralimia 3157 | . 2 ⊢ (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
10 | 5, 9 | syl 17 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ∀wral 3137 ∃wrex 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 |
This theorem is referenced by: (None) |
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