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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 1430 | . 2 ⊢ (∀x(φ → ψ) → ∀x∀x(φ → ψ)) | |
2 | hbe1 1381 | . 2 ⊢ (∃xψ → ∀x∃xψ) | |
3 | 19.8a 1479 | . . . 4 ⊢ (ψ → ∃xψ) | |
4 | 3 | imim2i 12 | . . 3 ⊢ ((φ → ψ) → (φ → ∃xψ)) |
5 | 4 | sps 1427 | . 2 ⊢ (∀x(φ → ψ) → (φ → ∃xψ)) |
6 | 1, 2, 5 | exlimdh 1484 | 1 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1240 ∃wex 1378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: eximi 1488 exbi 1492 eximdh 1499 19.29 1508 19.25 1514 alexim 1533 19.23t 1564 spimt 1621 equvini 1638 nfexd 1641 ax10oe 1675 sbcof2 1688 spsbim 1721 mor 1939 rexim 2407 elex22 2563 elex2 2564 vtoclegft 2619 spcimgft 2623 spcimegft 2625 spc2gv 2637 spc3gv 2639 ssoprab2 5503 bj-inf2vnlem1 9430 |
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