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Mirrors > Home > NFE Home > Th. List > vtoclri | GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.) |
Ref | Expression |
---|---|
vtoclri.1 | ⊢ (x = A → (φ ↔ ψ)) |
vtoclri.2 | ⊢ ∀x ∈ B φ |
Ref | Expression |
---|---|
vtoclri | ⊢ (A ∈ B → ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtoclri.1 | . 2 ⊢ (x = A → (φ ↔ ψ)) | |
2 | vtoclri.2 | . . 3 ⊢ ∀x ∈ B φ | |
3 | 2 | rspec 2678 | . 2 ⊢ (x ∈ B → φ) |
4 | 1, 3 | vtoclga 2920 | 1 ⊢ (A ∈ B → ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 ∀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-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-v 2861 |
This theorem is referenced by: (None) |
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