Theorem r19.22si 1275

Description: Inference quantifying both antecedent and consequent.

Hypothesis

Ref	Expression
r19.22si.1	⊢ (φ → ψ)

Assertion

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

Proof of Theorem r19.22si

Step	Hyp	Ref	Expression
1		r19.22si.1	. . 3 ⊢ (φ → ψ)
2	1	a1i 7	. 2 ⊢ (x ∈ A → (φ → ψ))
3	2	r19.22i 1273	1 ⊢ (∃x ∈ A φ → ∃x ∈ A ψ)

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

This theorem is referenced by:  r19.40 1301  abrexex 2912  unbnn2 3436  scott0 3542  aceq6b 3565  numthlem 3598  numthcor 3601  zorn2 3612  cflim 3704  climunii 4883  hlimunii 5143

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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