Theorem r19.22sdv 1279

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.)

Hypothesis

Ref	Expression
r19.22sdv.1	⊢ (φ → (ψ → χ))

Assertion

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

Distinct variable group(s):   φ,x

Proof of Theorem r19.22sdv

Step	Hyp	Ref	Expression
1		r19.22sdv.1	. . 3 ⊢ (φ → (ψ → χ))
2	1	a1d 14	. 2 ⊢ (φ → (x ∈ A → (ψ → χ)))
3	2	r19.22dv 1278	1 ⊢ (φ → (∃x ∈ A ψ → ∃x ∈ A χ))

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

This theorem is referenced by:  iunpw 2040  fvelima 2859  ssfi 3430  isfinite2 3437  hlimcaui 5141  occllem6 5185  projlem25 5217  projlem31 5223  osumlem4 5533

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

metamath.org