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| Mirrors > Home > MPE Home > Th. List > 19.31 | Structured version Visualization version GIF version | ||
| Description: Theorem 19.31 of [Margaris] p. 90. See 19.31v 1940 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) | 
| Ref | Expression | 
|---|---|
| 19.31.1 | ⊢ Ⅎ𝑥𝜓 | 
| Ref | Expression | 
|---|---|
| 19.31 | ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 19.31.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
| 2 | 1 | 19.32 2232 | . 2 ⊢ (∀𝑥(𝜓 ∨ 𝜑) ↔ (𝜓 ∨ ∀𝑥𝜑)) | 
| 3 | orcom 870 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑)) | |
| 4 | 3 | albii 1818 | . 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(𝜓 ∨ 𝜑)) | 
| 5 | orcom 870 | . 2 ⊢ ((∀𝑥𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ∀𝑥𝜑)) | |
| 6 | 2, 4, 5 | 3bitr4i 303 | 1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 847 ∀wal 1537 Ⅎwnf 1782 | 
| 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-or 848 df-ex 1779 df-nf 1783 | 
| This theorem is referenced by: 2eu3 2653 | 
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