|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > 19.44 | Structured version Visualization version GIF version | ||
| Description: Theorem 19.44 of [Margaris] p. 90. See 19.44v 1992 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) | 
| Ref | Expression | 
|---|---|
| 19.44.1 | ⊢ Ⅎ𝑥𝜓 | 
| Ref | Expression | 
|---|---|
| 19.44 | ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 19.43 1882 | . 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) | |
| 2 | 19.44.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 3 | 2 | 19.9 2205 | . . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓) | 
| 4 | 3 | orbi2i 913 | . 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓)) | 
| 5 | 1, 4 | bitri 275 | 1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 848 ∃wex 1779 Ⅎwnf 1783 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-or 849 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: eeorOLD 2336 | 
| Copyright terms: Public domain | W3C validator |