Theorem ralimia 2593

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 2592	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

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

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

This theorem is referenced by:  ralimiaa  2594  ralimi  2595  r19.12  2639  rr19.3v  2945  rr19.28v  2946  ffvresb  5810  f1mpt  5911  ixpf  6888  exmidontri2or  7460  peano2nnnn  8072  peano5nnnn  8111  peano5nni  9145  peano2nn  9154  serf0  11912  baspartn  14773  tridceq  16660