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Mirrors > Home > ILE Home > Th. List > ralbiia | GIF version |
Description: Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.) |
Ref | Expression |
---|---|
ralbiia.1 | ⊢ (x ∈ A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
ralbiia | ⊢ (∀x ∈ A φ ↔ ∀x ∈ A ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbiia.1 | . . 3 ⊢ (x ∈ A → (φ ↔ ψ)) | |
2 | 1 | pm5.74i 169 | . 2 ⊢ ((x ∈ A → φ) ↔ (x ∈ A → ψ)) |
3 | 2 | ralbii2 2328 | 1 ⊢ (∀x ∈ A φ ↔ ∀x ∈ A ψ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∈ wcel 1390 ∀wral 2300 |
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 |
This theorem depends on definitions: df-bi 110 df-ral 2305 |
This theorem is referenced by: poinxp 4352 soinxp 4353 seinxp 4354 dffun8 4872 funcnv3 4904 fncnv 4908 fnres 4958 fvreseq 5214 isoini2 5401 smores 5848 |
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