![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rexbii2 | Structured version Visualization version GIF version |
Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.) |
Ref | Expression |
---|---|
rexbii2.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)) |
Ref | Expression |
---|---|
rexbii2 | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexbii2.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)) | |
2 | 1 | exbii 1842 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 ∃wrex 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 |
This theorem depends on definitions: df-bi 206 df-ex 1774 df-rex 3063 |
This theorem is referenced by: rexbiia 3084 rexeqbii 3331 rexrab 3684 rexin 4231 rexdifpr 4653 rexdifsn 4789 reusv2lem4 5389 reusv2 5391 frpoind 6333 wfiOLD 6342 eldifsucnn 8658 frind 9740 rexuz2 12879 rexrp 12991 rexuz3 15291 infpn2 16842 efgrelexlemb 19655 cmpcov2 23204 cmpfi 23222 txkgen 23466 cubic 26685 madeval2 27684 sumdmdii 32092 bnj882 34392 bnj893 34394 heibor1 37134 eldmqsres 37611 prtlem100 38185 islmodfg 42266 limcrecl 44796 |
Copyright terms: Public domain | W3C validator |