Theorem reximdv2 2412

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)

Hypothesis

Ref	Expression
reximdv2.1	⊢ (φ → ((x ∈ A ∧ ψ) → (x ∈ B ∧ χ)))

Assertion

Ref	Expression
reximdv2	⊢ (φ → (∃x ∈ A ψ → ∃x ∈ B χ))

Distinct variable group:   φ,x

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

Proof of Theorem reximdv2

Step	Hyp	Ref	Expression
1		reximdv2.1	. . 3 ⊢ (φ → ((x ∈ A ∧ ψ) → (x ∈ B ∧ χ)))
2	1	eximdv 1757	. 2 ⊢ (φ → (∃x(x ∈ A ∧ ψ) → ∃x(x ∈ B ∧ χ)))
3		df-rex 2306	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ))
4		df-rex 2306	. 2 ⊢ (∃x ∈ B χ ↔ ∃x(x ∈ B ∧ χ))
5	2, 3, 4	3imtr4g 194	1 ⊢ (φ → (∃x ∈ A ψ → ∃x ∈ B χ))

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-rex 2306

This theorem is referenced by:  ssrexv  2999  ssimaex  5177