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Mirrors > Home > ILE Home > Th. List > rexeqbii | GIF version |
Description: Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
raleqbii.1 | ⊢ 𝐴 = 𝐵 |
raleqbii.2 | ⊢ (𝜓 ↔ 𝜒) |
Ref | Expression |
---|---|
rexeqbii | ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqbii.1 | . . . 4 ⊢ 𝐴 = 𝐵 | |
2 | 1 | eleq2i 2184 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
3 | raleqbii.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
4 | 2, 3 | anbi12i 455 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)) |
5 | 4 | rexbii2 2423 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 ∈ wcel 1465 ∃wrex 2394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-17 1491 ax-ial 1499 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-cleq 2110 df-clel 2113 df-rex 2399 |
This theorem is referenced by: exmidsbthrlem 13144 |
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