Theorem exintr 1113

Description: Introduce a conjunct in the scope of an existential quantifier.

Assertion

Ref	Expression
exintr	⊢ (∀x(φ → ψ) → (∃xφ → ∃x(φ ⋀ ψ)))

Proof of Theorem exintr

Step	Hyp	Ref	Expression
1		hba1 1000	. 2 ⊢ (∀x(φ → ψ) → ∀x∀x(φ → ψ))
2		ancl 294	. . 3 ⊢ ((φ → ψ) → (φ → (φ ⋀ ψ)))
3	2	a4s 981	. 2 ⊢ (∀x(φ → ψ) → (φ → (φ ⋀ ψ)))
4	1, 3	19.22d 1058	1 ⊢ (∀x(φ → ψ) → (∃xφ → ∃x(φ ⋀ ψ)))

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 951  ∃wex 977

This theorem is referenced by:  ceqsex 1825  r19.2z 2337  pwpw0 2460

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978

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