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Mirrors > Home > MPE Home > Th. List > r19.12 | Structured version Visualization version GIF version |
Description: Restricted quantifier version of 19.12 2335. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2379, ax-ext 2770. (Revised by Wolf Lammen, 17-Jun-2023.) |
Ref | Expression |
---|---|
r19.12 | ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3112 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑)) | |
2 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
3 | nfra1 3183 | . . . . 5 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑 | |
4 | 2, 3 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑) |
5 | 4 | nfex 2332 | . . 3 ⊢ Ⅎ𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑) |
6 | 1, 5 | nfxfr 1854 | . 2 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
7 | ax-1 6 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)) | |
8 | rsp 3170 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑)) | |
9 | 8 | com12 32 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 𝜑 → 𝜑)) |
10 | 9 | reximdv 3232 | . . 3 ⊢ (𝑦 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
11 | 7, 10 | sylcom 30 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝜑)) |
12 | 6, 11 | ralrimi 3180 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∃wex 1781 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-ral 3111 df-rex 3112 |
This theorem is referenced by: iuniin 4893 ucncn 22891 ftc1a 24640 heicant 35092 rngoid 35340 rngmgmbs4 35369 intimass 40355 intimag 40357 |
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