Theorem exintr 1614

Description: Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)

Assertion

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

Proof of Theorem exintr

Step	Hyp	Ref	Expression
1		exintrbi 1613	. 2 ⊢ (∀x(φ → ψ) → (∃xφ ↔ ∃x(φ ∧ ψ)))
2	1	biimpd 198	1 ⊢ (∀x(φ → ψ) → (∃xφ → ∃x(φ ∧ ψ)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is used by:  ceqsex  2894  r19.2z  3640  pwpw0  3856  pwsnALT  3883