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Mirrors > Home > NFE 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 2639 | . 2 ⊢ (∃y ∈ B φ ↔ ∃y ∈ B ψ) |
3 | 2 | rexbii 2639 | 1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x ∈ A ∃y ∈ B ψ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∃wrex 2615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-rex 2620 |
This theorem is referenced by: 3reeanv 2779 addccom 4406 addcass 4415 ncfinraise 4481 ncfinlower 4483 nnpweq 4523 dfxp2 5113 peano4nc 6150 sbth 6206 |
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