Theorem exintr 793

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 698	. 2 ⊢ (∀x(φ → ψ) → ∀x∀x(φ → ψ))
2		ancl 242	. . 3 ⊢ ((φ → ψ) → (φ → (φ ∧ ψ)))
3	2	a4s 682	. 2 ⊢ (∀x(φ → ψ) → (φ → (φ ∧ ψ)))
4	1, 3	19.22d 744	1 ⊢ (∀x(φ → ψ) → (∃xφ → ∃x(φ ∧ ψ)))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678

This theorem is referenced by:  ceqsex 1370  r19.2z 1766  pwpw0 1883

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

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