Theorem ralimia 2538

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)

Hypothesis

Ref	Expression
ralimia.1	⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))

Assertion

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

Proof of Theorem ralimia

Step	Hyp	Ref	Expression
1		ralimia.1	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
2	1	a2i 11	. 2 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜓))
3	2	ralimi2 2537	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2148  ∀wral 2455

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 1447  ax-gen 1449

This theorem depends on definitions:  df-bi 117  df-ral 2460

This theorem is referenced by:  ralimiaa  2539  ralimi  2540  r19.12  2583  rr19.3v  2876  rr19.28v  2877  ffvresb  5676  f1mpt  5767  ixpf  6715  exmidontri2or  7237  peano2nnnn  7847  peano5nnnn  7886  peano5nni  8916  peano2nn  8925  serf0  11351  baspartn  13330  tridceq  14575