Theorem ralbiia 2544

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 2540	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   ∈ wcel 2200  ∀wral 2508

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 1493  ax-gen 1495

This theorem depends on definitions:  df-bi 117  df-ral 2513

This theorem is referenced by:  frind  4443  poinxp  4788  soinxp  4789  seinxp  4790  dffun8  5346  funcnv3  5383  fncnv  5387  fnres  5440  fvreseq  5738  isoini2  5943  smores  6438  resixp  6880  pw1dc1  7076  finomni  7307  caucvgre  11492  xpscf  13380  mpodvdsmulf1o  15664  bj-charfundcALT  16172  cndcap  16427