Theorem reximdai 2723

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)

Hypotheses

Ref	Expression
reximdai.1	⊢ Ⅎxφ
reximdai.2	⊢ (φ → (x ∈ A → (ψ → χ)))

Assertion

Ref	Expression
reximdai	⊢ (φ → (∃x ∈ A ψ → ∃x ∈ A χ))

Proof of Theorem reximdai

Step	Hyp	Ref	Expression
1		reximdai.1	. . 3 ⊢ Ⅎxφ
2		reximdai.2	. . 3 ⊢ (φ → (x ∈ A → (ψ → χ)))
3	1, 2	ralrimi 2696	. 2 ⊢ (φ → ∀x ∈ A (ψ → χ))
4		rexim 2719	. 2 ⊢ (∀x ∈ A (ψ → χ) → (∃x ∈ A ψ → ∃x ∈ A χ))
5	3, 4	syl 15	1 ⊢ (φ → (∃x ∈ A ψ → ∃x ∈ A χ))

Syntax hints:   → wi 4  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by:  reximdvai  2725  chfnrn  5400