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Mirrors > Home > ILE Home > Th. List > 19.41v | GIF version |
Description: Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
19.41v | ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1537 | . 2 ⊢ (𝜓 → ∀𝑥𝜓) | |
2 | 1 | 19.41h 1696 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∃wex 1503 |
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 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: 19.41vv 1915 19.41vvv 1916 19.41vvvv 1917 exdistrv 1922 eeeanv 1945 gencbvex 2798 euxfrdc 2938 euind 2939 dfdif3 3260 r19.9rmv 3529 opabm 4298 eliunxp 4784 relop 4795 dmuni 4855 dmres 4946 dminss 5061 imainss 5062 ssrnres 5089 cnvresima 5136 resco 5151 rnco 5153 coass 5165 xpcom 5193 f11o 5513 fvelrnb 5584 rnoprab 5979 domen 6777 xpassen 6856 genpassl 7553 genpassu 7554 |
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