Theorem exintrbi 1633

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		pm4.71 389	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓)))
2	1	albii 1470	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑥(𝜑 ↔ (𝜑 ∧ 𝜓)))
3		exbi 1604	. 2 ⊢ (∀𝑥(𝜑 ↔ (𝜑 ∧ 𝜓)) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ 𝜓)))
4	2, 3	sylbi 121	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  exintr  1634