Theorem ralimdv 2600

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 276	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
3	2	ralimdva 2599	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∈ wcel 2202  ∀wral 2510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-gen 1497  ax-4 1558  ax-17 1574

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515

This theorem is referenced by:  poss  4395  sess1  4434  sess2  4435  riinint  4993  dffo4  5795  dffo5  5796  isoini2  5959  rdgivallem  6546  iinerm  6775  xpf1o  7029  exmidontriimlem3  7437  exmidontriim  7439  resqrexlemgt0  11580  cau3lem  11674  caubnd2  11677  climshftlemg  11862  climcau  11907  climcaucn  11911  serf0  11912  modfsummodlemstep  12017  bezoutlemmain  12568  ctinf  13050  strsetsid  13114  imasaddfnlemg  13396  islss4  14395  fiinbas  14772  baspartn  14773  lmtopcnp  14973  rescncf  15304  limcresi  15389  upgrwlkedg  16211  uspgr2wlkeq  16215  umgrwlknloop  16218