Theorem rexlimddv 2742

Description: Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)

Hypotheses

Ref	Expression
rexlimddv.1	⊢ (φ → ∃x ∈ A ψ)
rexlimddv.2	⊢ ((φ ∧ (x ∈ A ∧ ψ)) → χ)

Assertion

Ref	Expression
rexlimddv	⊢ (φ → χ)

Distinct variable groups:   φ,x   χ,x

Allowed substitution hints:   ψ(x)   A(x)

Proof of Theorem rexlimddv

Step	Hyp	Ref	Expression
1		rexlimddv.1	. 2 ⊢ (φ → ∃x ∈ A ψ)
2		rexlimddv.2	. . 3 ⊢ ((φ ∧ (x ∈ A ∧ ψ)) → χ)
3	2	rexlimdvaa 2739	. 2 ⊢ (φ → (∃x ∈ A ψ → χ))
4	1, 3	mpd 14	1 ⊢ (φ → χ)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∃wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619  df-rex 2620

This theorem is referenced by: (None)