Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2539 | . 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 104 Ⅎwnf 1436 ∀wral 2416 ∃wrex 2417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-ral 2421 df-rex 2422 |
This theorem is referenced by: r19.23v 2541 rexlimi 2542 rexlimd 2546 |
Copyright terms: Public domain | W3C validator |