Theorem r19.22dv2 1277

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90.

Hypothesis

Ref	Expression
r19.22dv2.1	⊢ (φ → ((x ∈ A ∧ ψ) → (x ∈ B ∧ χ)))

Assertion

Ref	Expression
r19.22dv2	⊢ (φ → (∃x ∈ A ψ → ∃x ∈ B χ))

Distinct variable group(s):   φ,x

Proof of Theorem r19.22dv2

Step	Hyp	Ref	Expression
1		r19.22dv2.1	. . 3 ⊢ (φ → ((x ∈ A ∧ ψ) → (x ∈ B ∧ χ)))
2	1	19.22dv 947	. 2 ⊢ (φ → (∃x(x ∈ A ∧ ψ) → ∃x(x ∈ B ∧ χ)))
3		df-rex 1206	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ))
4		df-rex 1206	. 2 ⊢ (∃x ∈ B χ ↔ ∃x(x ∈ B ∧ χ))
5	2, 3, 4	3imtr4g 426	1 ⊢ (φ → (∃x ∈ A ψ → ∃x ∈ B χ))

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  iunss1 2002  oaass 3163  zfregs 3491

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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