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Mirrors > Home > ILE Home > Th. List > r19.23 | GIF version |
Description: Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.) |
Ref | Expression |
---|---|
r19.23.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
r19.23 | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.23.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | r19.23t 2480 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 Ⅎwnf 1395 ∀wral 2360 ∃wrex 2361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-4 1446 ax-ial 1473 ax-i5r 1474 |
This theorem depends on definitions: df-bi 116 df-nf 1396 df-ral 2365 df-rex 2366 |
This theorem is referenced by: r19.23v 2482 rexlimi 2483 rexlimd 2487 |
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