Theorem r19.22dva 1280

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

Hypothesis

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

Assertion

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

Distinct variable group(s):   φ,x

Proof of Theorem r19.22dva

Step	Hyp	Ref	Expression
1		r19.22dva.1	. . 3 ⊢ ((φ ∧ x ∈ A) → (ψ → χ))
2	1	exp 291	. 2 ⊢ (φ → (x ∈ A → (ψ → χ)))
3	2	r19.22dv 1278	1 ⊢ (φ → (∃x ∈ A ψ → ∃x ∈ A χ))

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  arch 4521  uzwo3lem1 4614  projlem15 5207  projlem26 5218  projlem28 5220  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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