Theorem rexbii2 3081

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 3063	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-rex 3063	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
5	2, 3, 4	3bitr4i 303	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 207  df-ex 1782  df-rex 3063

This theorem is referenced by:  rexbiia  3083  rexeqbii  3311  rexrab  3643  rexin  4191  rexdifpr  4604  rexdifsn  4738  reusv2lem4  5338  reusv2  5340  frpoind  6300  eldifsucnn  8593  frind  9665  rexuz2  12840  rexrp  12956  rexuz3  15302  infpn2  16875  efgrelexlemb  19716  cmpcov2  23365  cmpfi  23383  txkgen  23627  cubic  26826  madeval2  27839  sumdmdii  32501  extdgfialglem1  33852  bnj882  35084  bnj893  35086  heibor1  38145  eldmqsres  38628  prtlem100  39319  islmodfg  43515  iuneq1i  45533  limcrecl  46077