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Theorem ralimia 2425
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
Hypothesis
Ref Expression
ralimia.1 (𝑥𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ralimia (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)

Proof of Theorem ralimia
StepHypRef Expression
1 ralimia.1 . . 3 (𝑥𝐴 → (𝜑𝜓))
21a2i 11 . 2 ((𝑥𝐴𝜑) → (𝑥𝐴𝜓))
32ralimi2 2424 1 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379
This theorem depends on definitions:  df-bi 115  df-ral 2354
This theorem is referenced by:  ralimiaa  2426  ralimi  2427  r19.12  2467  rr19.3v  2734  rr19.28v  2735  ffvresb  5360  f1mpt  5442  peano2nnnn  7083  peano5nnnn  7120  peano5nni  8109  peano2nn  8118  serif0  10327
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