![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rexbid2 | GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.) |
Ref | Expression |
---|---|
rexbid2.nf | ⊢ Ⅎ𝑥𝜑 |
rexbid2.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) |
Ref | Expression |
---|---|
rexbid2 | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexbid2.nf | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | rexbid2.1 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) | |
3 | 1, 2 | exbid 1593 | . 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))) |
4 | df-rex 2438 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
5 | df-rex 2438 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)) | |
6 | 3, 4, 5 | 3bitr4g 222 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 Ⅎwnf 1437 ∃wex 1469 ∈ wcel 2125 ∃wrex 2433 |
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 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1487 ax-ial 1511 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-rex 2438 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |