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Mirrors > Home > ILE Home > Th. List > exim | GIF version |
Description: Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
Ref | Expression |
---|---|
exim | ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hba1 1540 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥∀𝑥(𝜑 → 𝜓)) | |
2 | hbe1 1495 | . 2 ⊢ (∃𝑥𝜓 → ∀𝑥∃𝑥𝜓) | |
3 | 19.8a 1590 | . . . 4 ⊢ (𝜓 → ∃𝑥𝜓) | |
4 | 3 | imim2i 12 | . . 3 ⊢ ((𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)) |
5 | 4 | sps 1537 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)) |
6 | 1, 2, 5 | exlimdh 1596 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1351 ∃wex 1492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: eximi 1600 exbi 1604 eximdh 1611 19.29 1620 19.25 1626 alexim 1645 19.23t 1677 spimt 1736 equvini 1758 nfexd 1761 ax10oe 1797 sbcof2 1810 spsbim 1843 nf5-1 2024 mor 2068 rexim 2571 elex22 2752 elex2 2753 vtoclegft 2809 spcimgft 2813 spcimegft 2815 spc2gv 2828 spc3gv 2830 ssoprab2 5924 bj-inf2vnlem1 14344 |
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