Theorem r19.22d 1276

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

Hypotheses

Ref	Expression
r19.22d.1	⊢ (φ → ∀xφ)
r19.22d.2	⊢ (φ → (x ∈ A → (ψ → χ)))

Assertion

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

Proof of Theorem r19.22d

Step	Hyp	Ref	Expression
1		r19.22d.1	. . 3 ⊢ (φ → ∀xφ)
2		r19.22d.2	. . 3 ⊢ (φ → (x ∈ A → (ψ → χ)))
3	1, 2	r19.21ai 1258	. 2 ⊢ (φ → ∀x ∈ A (ψ → χ))
4		r19.22 1272	. 2 ⊢ (∀x ∈ A (ψ → χ) → (∃x ∈ A ψ → ∃x ∈ A χ))
5	3, 4	syl 12	1 ⊢ (φ → (∃x ∈ A ψ → ∃x ∈ A χ))

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202

This theorem is referenced by:  r19.22dv 1278  ss2iun 2005  chfnrn 2885  tz7.49 2997  r1tr 3498

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

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

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