Theorem rexim 2407

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.)

Assertion

Ref	Expression
rexim	⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ))

Proof of Theorem rexim

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))

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