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Mirrors > Home > MPE Home > Th. List > Mathboxes > 19.9d2r | Structured version Visualization version GIF version |
Description: A deduction version of one direction of 19.9 2197 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
Ref | Expression |
---|---|
19.9d2r.1 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
19.9d2r.2 | ⊢ (𝜑 → Ⅎ𝑦𝜓) |
19.9d2r.3 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) |
Ref | Expression |
---|---|
19.9d2r | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1916 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | 19.9d2r.1 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | 19.9d2r.2 | . 2 ⊢ (𝜑 → Ⅎ𝑦𝜓) | |
4 | 19.9d2r.3 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) | |
5 | 1, 2, 3, 4 | 19.9d2rf 31108 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1784 ∃wrex 3070 |
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 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-11 2153 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1781 df-nf 1785 df-rex 3071 |
This theorem is referenced by: xrofsup 31377 |
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