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Mirrors > Home > MPE Home > Th. List > 19.27 | Structured version Visualization version GIF version |
Description: Theorem 19.27 of [Margaris] p. 90. See 19.27v 2094 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
19.27.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
19.27 | ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.26 1972 | . 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) | |
2 | 19.27.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | 19.3 2243 | . . 3 ⊢ (∀𝑥𝜓 ↔ 𝜓) |
4 | 3 | anbi2i 616 | . 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓)) |
5 | 1, 4 | bitri 267 | 1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∀wal 1654 Ⅎwnf 1882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-12 2220 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1879 df-nf 1883 |
This theorem is referenced by: aaan 2363 |
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