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Mirrors > Home > NFE Home > Th. List > 2ralbidv | GIF version |
Description: Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.) |
Ref | Expression |
---|---|
2ralbidv.1 | ⊢ (φ → (ψ ↔ χ)) |
Ref | Expression |
---|---|
2ralbidv | ⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2ralbidv.1 | . . 3 ⊢ (φ → (ψ ↔ χ)) | |
2 | 1 | ralbidv 2634 | . 2 ⊢ (φ → (∀y ∈ B ψ ↔ ∀y ∈ B χ)) |
3 | 2 | ralbidv 2634 | 1 ⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∀wral 2614 |
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-ex 1542 df-nf 1545 df-ral 2619 |
This theorem is referenced by: cbvral3v 2845 isoeq1 5482 isoeq2 5483 isoeq3 5484 trd 5921 extd 5923 symd 5924 trrd 5925 antird 5928 antid 5929 connexrd 5930 connexd 5931 iserd 5942 |
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