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Mirrors > Home > MPE Home > Th. List > r19.36v | Structured version Visualization version GIF version |
Description: Restricted quantifier version of one direction of 19.36 2230. (The other direction holds iff 𝐴 is nonempty, see r19.36zv 4410.) (Contributed by NM, 22-Oct-2003.) |
Ref | Expression |
---|---|
r19.36v | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.35 3295 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | id 22 | . . . 4 ⊢ (𝜓 → 𝜓) | |
3 | 2 | rexlimivw 3241 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝜓) |
4 | 3 | imim2i 16 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
5 | 1, 4 | sylbi 220 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-ral 3111 df-rex 3112 |
This theorem is referenced by: iinss 4943 uniimadom 9955 hashgt12el 13779 |
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