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Mirrors > Home > ILE Home > Th. List > r19.43 | GIF version |
Description: Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.) |
Ref | Expression |
---|---|
r19.43 | ⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2306 | . . . 4 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ ∃x(x ∈ A ∧ (φ ∨ ψ))) | |
2 | andi 730 | . . . . 5 ⊢ ((x ∈ A ∧ (φ ∨ ψ)) ↔ ((x ∈ A ∧ φ) ∨ (x ∈ A ∧ ψ))) | |
3 | 2 | exbii 1493 | . . . 4 ⊢ (∃x(x ∈ A ∧ (φ ∨ ψ)) ↔ ∃x((x ∈ A ∧ φ) ∨ (x ∈ A ∧ ψ))) |
4 | 1, 3 | bitri 173 | . . 3 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ ∃x((x ∈ A ∧ φ) ∨ (x ∈ A ∧ ψ))) |
5 | 19.43 1516 | . . 3 ⊢ (∃x((x ∈ A ∧ φ) ∨ (x ∈ A ∧ ψ)) ↔ (∃x(x ∈ A ∧ φ) ∨ ∃x(x ∈ A ∧ ψ))) | |
6 | 4, 5 | bitri 173 | . 2 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x(x ∈ A ∧ φ) ∨ ∃x(x ∈ A ∧ ψ))) |
7 | df-rex 2306 | . . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ)) | |
8 | df-rex 2306 | . . 3 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ)) | |
9 | 7, 8 | orbi12i 680 | . 2 ⊢ ((∃x ∈ A φ ∨ ∃x ∈ A ψ) ↔ (∃x(x ∈ A ∧ φ) ∨ ∃x(x ∈ A ∧ ψ))) |
10 | 6, 9 | bitr4i 176 | 1 ⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ 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-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-rex 2306 |
This theorem is referenced by: r19.44av 2463 r19.45av 2464 r19.45mv 3309 iunun 3725 ltexprlemloc 6581 |
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