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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 19.9d2rf | Structured version Visualization version GIF version |
Description: A deduction version of one direction of 19.9 2247 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.) |
Ref | Expression |
---|---|
19.9d2rf.0 | ⊢ Ⅎ𝑦𝜑 |
19.9d2rf.1 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
19.9d2rf.2 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
19.9d2rf.3 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) |
Ref | Expression |
---|---|
19.9d2rf | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.9d2rf.3 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) | |
2 | rexex 3210 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑥∃𝑦 ∈ 𝐵 𝜓) | |
3 | rexex 3210 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝜓 → ∃𝑦𝜓) | |
4 | 3 | eximi 1933 | . . . 4 ⊢ (∃𝑥∃𝑦 ∈ 𝐵 𝜓 → ∃𝑥∃𝑦𝜓) |
5 | 1, 2, 4 | 3syl 18 | . . 3 ⊢ (𝜑 → ∃𝑥∃𝑦𝜓) |
6 | 19.9d2rf.0 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
7 | 19.9d2rf.1 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
8 | 6, 7 | nfexd 2361 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
9 | 8 | 19.9d 2244 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓)) |
10 | 5, 9 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑦𝜓) |
11 | 19.9d2rf.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
12 | 11 | 19.9d 2244 | . 2 ⊢ (𝜑 → (∃𝑦𝜓 → 𝜓)) |
13 | 10, 12 | mpd 15 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1878 Ⅎwnf 1882 ∃wrex 3118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-10 2192 ax-11 2207 ax-12 2220 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-ex 1879 df-nf 1883 df-rex 3123 |
This theorem is referenced by: 19.9d2r 29870 xrofsup 30076 |
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