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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 2314. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2365, ax-ext 2697. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.) |
Ref | Expression |
---|---|
r19.12 | ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3065 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑)) | |
2 | nfv 1909 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
3 | nfra1 3275 | . . . . 5 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑 | |
4 | 2, 3 | nfan 1894 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑) |
5 | 4 | nfex 2311 | . . 3 ⊢ Ⅎ𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝜑) |
6 | 1, 5 | nfxfr 1847 | . 2 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
7 | rsp 3238 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑)) | |
8 | 7 | com12 32 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 𝜑 → 𝜑)) |
9 | 8 | reximdv 3164 | . . 3 ⊢ (𝑦 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)) |
10 | 9 | com12 32 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝜑)) |
11 | 6, 10 | ralrimi 3248 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-11 2146 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-ral 3056 df-rex 3065 |
This theorem is referenced by: iuniin 5002 ucncn 24141 ftc1a 25923 heicant 37034 rngoid 37281 rngmgmbs4 37310 intimass 42962 intimag 42964 |
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