![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rmobiia | GIF version |
Description: Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
rmobiia.1 | ⊢ (x ∈ A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
rmobiia | ⊢ (∃*x ∈ A φ ↔ ∃*x ∈ A ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmobiia.1 | . . . 4 ⊢ (x ∈ A → (φ ↔ ψ)) | |
2 | 1 | pm5.32i 427 | . . 3 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ A ∧ ψ)) |
3 | 2 | mobii 1934 | . 2 ⊢ (∃*x(x ∈ A ∧ φ) ↔ ∃*x(x ∈ A ∧ ψ)) |
4 | df-rmo 2308 | . 2 ⊢ (∃*x ∈ A φ ↔ ∃*x(x ∈ A ∧ φ)) | |
5 | df-rmo 2308 | . 2 ⊢ (∃*x ∈ A ψ ↔ ∃*x(x ∈ A ∧ ψ)) | |
6 | 3, 4, 5 | 3bitr4i 201 | 1 ⊢ (∃*x ∈ A φ ↔ ∃*x ∈ A ψ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1390 ∃*wmo 1898 ∃*wrmo 2303 |
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 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-eu 1900 df-mo 1901 df-rmo 2308 |
This theorem is referenced by: rmobii 2494 |
Copyright terms: Public domain | W3C validator |