Theorem ralbiia 2511

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 2507	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 105   ∈ wcel 2167  ∀wral 2475

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 1461  ax-gen 1463

This theorem depends on definitions:  df-bi 117  df-ral 2480

This theorem is referenced by:  frind  4387  poinxp  4732  soinxp  4733  seinxp  4734  dffun8  5286  funcnv3  5320  fncnv  5324  fnres  5374  fvreseq  5665  isoini2  5866  smores  6350  resixp  6792  pw1dc1  6975  finomni  7206  caucvgre  11146  xpscf  12990  mpodvdsmulf1o  15226  bj-charfundcALT  15455  cndcap  15703