|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > r19.23 | Structured version Visualization version GIF version | ||
| Description: Restricted quantifier version of 19.23 2210. See r19.23v 3182 for a version requiring fewer axioms. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.) | 
| Ref | Expression | 
|---|---|
| r19.23.1 | ⊢ Ⅎ𝑥𝜓 | 
| Ref | Expression | 
|---|---|
| r19.23 | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | r19.23.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 2 | r19.23t 3254 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 Ⅎwnf 1782 ∀wral 3060 ∃wrex 3069 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-ral 3061 df-rex 3070 | 
| This theorem is referenced by: rexlimi 3258 iunssf 5043 ralxp3f 8163 | 
| Copyright terms: Public domain | W3C validator |