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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 2597 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 Ⅎwnf 1471 ∀wral 2468 ∃wrex 2469 |
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-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-ral 2473 df-rex 2474 |
This theorem is referenced by: r19.23v 2599 rexlimi 2600 rexlimd 2604 |
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