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Mirrors > Home > MPE Home > Th. List > 19.24 | Structured version Visualization version GIF version |
Description: Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
Ref | Expression |
---|---|
19.24 | ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.2 1986 | . . 3 ⊢ (∀𝑥𝜓 → ∃𝑥𝜓) | |
2 | 1 | imim2i 16 | . 2 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
3 | 19.35 1884 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
4 | 2, 3 | sylibr 237 | 1 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-6 1975 |
This theorem depends on definitions: df-bi 210 df-ex 1787 |
This theorem is referenced by: (None) |
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