Theorem ralbiia 2546

Description: Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)

Hypothesis

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

Assertion

Ref	Expression
ralbiia	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)

Proof of Theorem ralbiia

Step	Hyp	Ref	Expression
1		ralbiia.1	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
2	1	pm5.74i 180	. 2 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐴 → 𝜓))
3	2	ralbii2 2542	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   ∈ 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:  frind  4449  poinxp  4795  soinxp  4796  seinxp  4797  dffun8  5354  funcnv3  5392  fncnv  5396  fnres  5449  fvreseq  5750  isoini2  5960  smores  6458  resixp  6902  pw1dc1  7106  finomni  7339  caucvgre  11559  xpscf  13448  mpodvdsmulf1o  15733  bj-charfundcALT  16455  cndcap  16715