Theorem exintrbi 1613

Description: Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)

Assertion

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

Proof of Theorem exintrbi

Step	Hyp	Ref	Expression
1		pm4.71 611	. . 3 ⊢ ((φ → ψ) ↔ (φ ↔ (φ ∧ ψ)))
2	1	albii 1566	. 2 ⊢ (∀x(φ → ψ) ↔ ∀x(φ ↔ (φ ∧ ψ)))
3		exbi 1581	. 2 ⊢ (∀x(φ ↔ (φ ∧ ψ)) → (∃xφ ↔ ∃x(φ ∧ ψ)))
4	2, 3	sylbi 187	1 ⊢ (∀x(φ → ψ) → (∃xφ ↔ ∃x(φ ∧ ψ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

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

This theorem is referenced by:  exintr  1614