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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 1939 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 2231 | . 2 ⊢ (∀𝑥(𝜓 ∨ 𝜑) ↔ (𝜓 ∨ ∀𝑥𝜑)) |
3 | orcom 870 | . . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑)) | |
4 | 3 | albii 1816 | . 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 1535 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-or 848 df-ex 1777 df-nf 1781 |
This theorem is referenced by: 2eu3 2652 |
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