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Mirrors > Home > ILE Home > Th. List > rexim | GIF version |
Description: Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
rexim | ⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2305 | . . . 4 ⊢ (∀x ∈ A (φ → ψ) ↔ ∀x(x ∈ A → (φ → ψ))) | |
2 | simpl 102 | . . . . . . 7 ⊢ ((x ∈ A ∧ φ) → x ∈ A) | |
3 | 2 | a1i 9 | . . . . . 6 ⊢ ((x ∈ A → (φ → ψ)) → ((x ∈ A ∧ φ) → x ∈ A)) |
4 | pm3.31 249 | . . . . . 6 ⊢ ((x ∈ A → (φ → ψ)) → ((x ∈ A ∧ φ) → ψ)) | |
5 | 3, 4 | jcad 291 | . . . . 5 ⊢ ((x ∈ A → (φ → ψ)) → ((x ∈ A ∧ φ) → (x ∈ A ∧ ψ))) |
6 | 5 | alimi 1341 | . . . 4 ⊢ (∀x(x ∈ A → (φ → ψ)) → ∀x((x ∈ A ∧ φ) → (x ∈ A ∧ ψ))) |
7 | 1, 6 | sylbi 114 | . . 3 ⊢ (∀x ∈ A (φ → ψ) → ∀x((x ∈ A ∧ φ) → (x ∈ A ∧ ψ))) |
8 | exim 1487 | . . 3 ⊢ (∀x((x ∈ A ∧ φ) → (x ∈ A ∧ ψ)) → (∃x(x ∈ A ∧ φ) → ∃x(x ∈ A ∧ ψ))) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (∀x ∈ A (φ → ψ) → (∃x(x ∈ A ∧ φ) → ∃x(x ∈ A ∧ ψ))) |
10 | df-rex 2306 | . 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ)) | |
11 | df-rex 2306 | . 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ)) | |
12 | 9, 10, 11 | 3imtr4g 194 | 1 ⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1240 ∃wex 1378 ∈ wcel 1390 ∀wral 2300 ∃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-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-ral 2305 df-rex 2306 |
This theorem is referenced by: reximia 2408 reximdai 2411 r19.29 2444 reupick2 3217 ss2iun 3663 chfnrn 5221 |
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