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Mirrors > Home > MPE Home > Th. List > Mathboxes > r19.36vf | Structured version Visualization version GIF version |
Description: Restricted quantifier version of one direction of 19.36 2222. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
r19.36vf.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
r19.36vf | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.35 3107 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | r19.36vf.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | idd 24 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜓 → 𝜓)) | |
4 | 2, 3 | rexlimi 3238 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝜓) |
5 | 4 | imim2i 16 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
6 | 1, 5 | sylbi 216 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1784 ∈ wcel 2105 ∀wral 3061 ∃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-12 2170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1781 df-nf 1785 df-ral 3062 df-rex 3071 |
This theorem is referenced by: iinssf 43008 |
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