Theorem r19.22i 1273

Description: Inference quantifying both antecedent and consequent.

Hypothesis

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

Assertion

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

Proof of Theorem r19.22i

Step	Hyp	Ref	Expression
1		r19.22 1272	. 2 ⊢ (∀x ∈ A (φ → ψ) → (∃x ∈ A φ → ∃x ∈ A ψ))
2		r19.22i.1	. 2 ⊢ (x ∈ A → (φ → ψ))
3	1, 2	mprg 1249	1 ⊢ (∃x ∈ A φ → ∃x ∈ A ψ)

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

This theorem is referenced by:  r19.22si 1275  iunpw 2040  tfrlem8 2956  tz7.49c 2998  abianfp 3000  trcl 3489  rankwflem 3509  zqt 4632  projlem16 5208  chrelat2 5758

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

metamath.org