Theorem rexlimd 2736

Description: Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Hypotheses

Ref	Expression
rexlimd.1	⊢ Ⅎxφ
rexlimd.2	⊢ Ⅎxχ
rexlimd.3	⊢ (φ → (x ∈ A → (ψ → χ)))

Assertion

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

Proof of Theorem rexlimd

Step	Hyp	Ref	Expression
1		rexlimd.1	. . 3 ⊢ Ⅎxφ
2		rexlimd.3	. . 3 ⊢ (φ → (x ∈ A → (ψ → χ)))
3	1, 2	ralrimi 2696	. 2 ⊢ (φ → ∀x ∈ A (ψ → χ))
4		rexlimd.2	. . 3 ⊢ Ⅎxχ
5	4	r19.23 2730	. 2 ⊢ (∀x ∈ A (ψ → χ) ↔ (∃x ∈ A ψ → χ))
6	3, 5	sylib 188	1 ⊢ (φ → (∃x ∈ A ψ → χ))

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616

This proof depends on 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-6 1729  ax-11 1746

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

This theorem is used by:  rexlimdv  2738  fun11iun  5306  ffnfv  5428