Theorem rexbii2 3245

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 1844	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
3		df-rex 3144	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-rex 3144	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
5	2, 3, 4	3bitr4i 305	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1776   ∈ wcel 2110  ∃wrex 3139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 209  df-ex 1777  df-rex 3144

This theorem is referenced by:  rexbiia  3246  rexbii  3247  rexeqbii  3326  rexrab  3687  rexin  4216  rexdifpr  4592  rexdifsn  4721  reusv2lem4  5294  reusv2  5296  wfi  6176  rexuz2  12293  rexrp  12404  rexuz3  14702  infpn2  16243  efgrelexlemb  18870  cmpcov2  21992  cmpfi  22010  txkgen  22254  cubic  25421  sumdmdii  30186  bnj882  32193  bnj893  32195  frpoind  33075  frind  33080  madeval2  33285  heibor1  35082  eldmqsres  35537  prtlem100  35989  islmodfg  39662  limcrecl  41902