Theorem rexbiia 2565

Description: Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)

Hypothesis

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

Assertion

Ref	Expression
rexbiia	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓)

Proof of Theorem rexbiia

Step	Hyp	Ref	Expression
1		ralbiia.1	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
2	1	pm5.32i 458	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
3	2	rexbii2 2561	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   ∈ wcel 2209  ∃wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-rex 2534

This theorem is used by:  2rexbiia  2566  ceqsrexbv  2957  reu8  3022  reldm  6420  djur  7409  prarloclem3  7864  suplocexprlemell  8080  recexgt0  8908  fsum3  12154  fprodseq  12350  even2n  12641  znf1o  14986  lmres  15349  reeff1o  15874  ioocosf1o  15955