Theorem rexim 2468

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 2365	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
2		simpl 108	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴)
3	2	a1i 9	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴))
4		pm3.31 259	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
5	3, 4	jcad 302	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)))
6	5	alimi 1390	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)))
7	1, 6	sylbi 120	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)))
8		exim 1536	. . 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 2366	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
11		df-rex 2366	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
12	9, 10, 11	3imtr4g 204	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1288  ∃wex 1427   ∈ wcel 1439  ∀wral 2360  ∃wrex 2361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-ial 1473

This theorem depends on definitions:  df-bi 116  df-ral 2365  df-rex 2366

This theorem is referenced by:  reximia  2469  reximdai  2472  r19.29  2509  reupick2  3288  ss2iun  3753  chfnrn  5426