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Mirrors > Home > ILE Home > Th. List > r19.41v | GIF version |
Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.) |
Ref | Expression |
---|---|
r19.41v | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | r19.41 2465 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∃wrex 2307 |
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 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-rex 2312 |
This theorem is referenced by: r19.42v 2467 3reeanv 2480 reuind 2744 iuncom4 3664 dfiun2g 3689 iunxiun 3736 inuni 3909 xpiundi 4398 xpiundir 4399 imaco 4826 coiun 4830 abrexco 5398 imaiun 5399 isoini 5457 rexrnmpt2 5616 genpassl 6622 genpassu 6623 4fvwrd4 8997 |
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