Theorem rexbii2 3174

Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)

Hypothesis

Ref	Expression
rexbii2.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))

Assertion

Ref	Expression
rexbii2	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)

Proof of Theorem rexbii2

Step	Hyp	Ref	Expression
1		rexbii2.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))
2	1	exbii 1850	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
3		df-rex 3077	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-rex 3077	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
5	2, 3, 4	3bitr4i 307	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1782   ∈ wcel 2112  ∃wrex 3072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 210  df-ex 1783  df-rex 3077

This theorem is referenced by:  rexbiia  3175  rexeqbii  3251  rexrab  3611  rexin  4145  rexdifpr  4553  rexdifsn  4682  reusv2lem4  5268  reusv2  5270  wfi  6157  rexuz2  12329  rexrp  12441  rexuz3  14746  infpn2  16294  efgrelexlemb  18933  cmpcov2  22080  cmpfi  22098  txkgen  22342  cubic  25524  sumdmdii  30287  bnj882  32416  bnj893  32418  frpoind  33317  frind  33325  madeval2  33587  heibor1  35518  eldmqsres  35973  prtlem100  36425  islmodfg  40376  limcrecl  42627