Theorem r19.29 2755

Description: Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
r19.29	⊢ ((∀x ∈ A φ ∧ ∃x ∈ A ψ) → ∃x ∈ A (φ ∧ ψ))

Proof of Theorem r19.29

Step	Hyp	Ref	Expression
1		pm3.2 434	. . . 4 ⊢ (φ → (ψ → (φ ∧ ψ)))
2	1	ralimi 2690	. . 3 ⊢ (∀x ∈ A φ → ∀x ∈ A (ψ → (φ ∧ ψ)))
3		rexim 2719	. . 3 ⊢ (∀x ∈ A (ψ → (φ ∧ ψ)) → (∃x ∈ A ψ → ∃x ∈ A (φ ∧ ψ)))
4	2, 3	syl 15	. 2 ⊢ (∀x ∈ A φ → (∃x ∈ A ψ → ∃x ∈ A (φ ∧ ψ)))
5	4	imp 418	1 ⊢ ((∀x ∈ A φ ∧ ∃x ∈ A ψ) → ∃x ∈ A (φ ∧ ψ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wral 2615  ∃wrex 2616

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  df-ral 2620  df-rex 2621

This theorem is used by:  r19.29r  2756  fun11iun  5306  fmpt  5693