Theorem r19.22dv 1278

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90.

Hypothesis

Ref	Expression
r19.22dv.1	⊢ (φ → (x ∈ A → (ψ → χ)))

Assertion

Ref	Expression
r19.22dv	⊢ (φ → (∃x ∈ A ψ → ∃x ∈ A χ))

Distinct variable group(s):   φ,x

Proof of Theorem r19.22dv

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (φ → ∀xφ)
2		r19.22dv.1	. 2 ⊢ (φ → (x ∈ A → (ψ → χ)))
3	1, 2	r19.22d 1276	1 ⊢ (φ → (∃x ∈ A ψ → ∃x ∈ A χ))

Syntax hints:   → wi 2   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  r19.22sdv 1279  r19.22dva 1280  r19.12 1281  wefrc 2195  isomin 2937  isofrlem 2939  oaordex 3160  r1pwcl 3530  atcvat4 5775  mdsymlem2 5777  mdsymlem3 5778  sumdmdi 5785

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

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