Theorem 2rexbiia 3224

Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)

Hypothesis

Ref	Expression
2rexbiia.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
2rexbiia	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦)   𝐴(𝑥)   𝐵(𝑥, 𝑦)

Proof of Theorem 2rexbiia

Step	Hyp	Ref	Expression
1		2rexbiia.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 ↔ 𝜓))
2	1	rexbidva 3185	. 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))
3	2	rexbiia 3108	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  ∃wrex 3087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-rex 3088

This theorem is used by:  reu3op  6295  opreu2reurex  6297  cnref1o  13113  mndpfoOLD  18949  wspthsnwspthsnon  30505  mdsymlem8  33012  xlt2addrd  33351  elunirnmbfm  34885  satfv0  36123  fmla0xp  36148  icoreelrnab  38277  fimgmcyclem  43597  rrx2xpref1o  49829