![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > r19.45mv | GIF version |
Description: Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
Ref | Expression |
---|---|
r19.45mv | ⊢ (∃x x ∈ A → (∃x ∈ A (φ ∨ ψ) ↔ (φ ∨ ∃x ∈ A ψ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.9rmv 3307 | . . 3 ⊢ (∃x x ∈ A → (φ ↔ ∃x ∈ A φ)) | |
2 | 1 | orbi1d 704 | . 2 ⊢ (∃x x ∈ A → ((φ ∨ ∃x ∈ A ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ))) |
3 | r19.43 2462 | . 2 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ)) | |
4 | 2, 3 | syl6rbbr 188 | 1 ⊢ (∃x x ∈ A → (∃x ∈ A (φ ∨ ψ) ↔ (φ ∨ ∃x ∈ A ψ))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∨ wo 628 ∃wex 1378 ∈ wcel 1390 ∃wrex 2301 |
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-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-cleq 2030 df-clel 2033 df-rex 2306 |
This theorem is referenced by: ltexprlemloc 6581 |
Copyright terms: Public domain | W3C validator |