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Mirrors > Home > MPE Home > Th. List > 19.45 | Structured version Visualization version GIF version |
Description: Theorem 19.45 of [Margaris] p. 90. See 19.45v 1900 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) |
Ref | Expression |
---|---|
19.45.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.45 | ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.43 1799 | . 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) | |
2 | 19.45.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | 19.9 2060 | . . 3 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
4 | 3 | orbi1i 541 | . 2 ⊢ ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓)) |
5 | 1, 4 | bitri 263 | 1 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 ∃wex 1695 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-ex 1696 df-nf 1701 |
This theorem is referenced by: eeor 2157 |
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