Theorem ralimdv 2382

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)

Hypothesis

Ref	Expression
ralimdv.1	⊢ (φ → (ψ → χ))

Assertion

Ref	Expression
ralimdv	⊢ (φ → (∀x ∈ A ψ → ∀x ∈ A χ))

Distinct variable group:   φ,x

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

Proof of Theorem ralimdv

Step	Hyp	Ref	Expression
1		ralimdv.1	. . 3 ⊢ (φ → (ψ → χ))
2	1	adantr 261	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ → χ))
3	2	ralimdva 2381	1 ⊢ (φ → (∀x ∈ A ψ → ∀x ∈ A χ))

Syntax hints:   → wi 4   ∈ wcel 1390  ∀wral 2300

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-4 1397  ax-17 1416

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305

This theorem is referenced by:  poss  4026  sess1  4059  sess2  4060  riinint  4536  dffo4  5258  dffo5  5259  isoini2  5401  rdgivallem  5908  iinerm  6114