Theorem ralimdv 2443

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 271	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
3	2	ralimdva 2442	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∈ wcel 1439  ∀wral 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-4 1446  ax-17 1465

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-ral 2365

This theorem is referenced by:  poss  4134  sess1  4173  sess2  4174  riinint  4707  dffo4  5461  dffo5  5462  isoini2  5612  rdgivallem  6160  iinerm  6378  xpf1o  6614  resqrexlemgt0  10514  cau3lem  10608  caubnd2  10611  climshftlemg  10751  climcau  10797  climcaucn  10801  serf0  10802  modfsummodlemstep  10912  bezoutlemmain  11326  strsetsid  11588  fiinbas  11808  baspartn  11809  rescncf  11910