Theorem exintr 1525

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 1524	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ 𝜓)))
2	1	biimpd 132	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  ceqsex  2592  r19.2m  3309