Theorem rexlimd2 2737

Description: Version of rexlimd 2736 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
rexlimd2.1	⊢ Ⅎxφ
rexlimd2.2	⊢ (φ → Ⅎxχ)
rexlimd2.3	⊢ (φ → (x ∈ A → (ψ → χ)))

Assertion

Ref	Expression
rexlimd2	⊢ (φ → (∃x ∈ A ψ → χ))

Proof of Theorem rexlimd2

Step	Hyp	Ref	Expression
1		rexlimd2.1	. . 3 ⊢ Ⅎxφ
2		rexlimd2.3	. . 3 ⊢ (φ → (x ∈ A → (ψ → χ)))
3	1, 2	ralrimi 2696	. 2 ⊢ (φ → ∀x ∈ A (ψ → χ))
4		rexlimd2.2	. . 3 ⊢ (φ → Ⅎxχ)
5		r19.23t 2729	. . 3 ⊢ (Ⅎxχ → (∀x ∈ A (ψ → χ) ↔ (∃x ∈ A ψ → χ)))
6	4, 5	syl 15	. 2 ⊢ (φ → (∀x ∈ A (ψ → χ) ↔ (∃x ∈ A ψ → χ)))
7	3, 6	mpbid 201	1 ⊢ (φ → (∃x ∈ A ψ → χ))

Syntax hints:   → wi 4   ↔ wb 176  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-6 1729  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: (None)