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Mirrors > Home > ILE Home > Th. List > 2rexbii | GIF version |
Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.) |
Ref | Expression |
---|---|
ralbii.1 | ⊢ (φ ↔ ψ) |
Ref | Expression |
---|---|
2rexbii | ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbii.1 | . . 3 ⊢ (φ ↔ ψ) | |
2 | 1 | rexbii 2325 | . 2 ⊢ (∃y ∈ B φ ↔ ∃y ∈ B ψ) |
3 | 2 | rexbii 2325 | 1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∃wrex 2301 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-rex 2306 |
This theorem is referenced by: 3reeanv 2474 4fvwrd4 8767 |
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