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Mirrors > Home > ILE Home > Th. List > 19.23t | GIF version |
Description: Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
Ref | Expression |
---|---|
19.23t | ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exim 1579 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.9t 1622 | . . . 4 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓)) | |
3 | 2 | biimpd 143 | . . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓)) |
4 | 1, 3 | syl9r 73 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓))) |
5 | nfr 1499 | . . . 4 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓)) | |
6 | 5 | imim2d 54 | . . 3 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) |
7 | 19.38 1655 | . . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) | |
8 | 6, 7 | syl6 33 | . 2 ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))) |
9 | 4, 8 | impbid 128 | 1 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1330 Ⅎwnf 1437 ∃wex 1469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-nf 1438 |
This theorem is referenced by: 19.23 1657 r19.23t 2542 ceqsalt 2715 vtoclgft 2739 sbciegft 2943 |
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