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Mirrors > Home > MPE Home > Th. List > vtoclri | Structured version Visualization version 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 3157 | . 2 ⊢ (𝑥 ∈ 𝐵 → 𝜑) |
4 | 1, 3 | vtoclga 3493 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 ∀wral 3088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-nf 1747 df-cleq 2771 df-clel 2846 df-ral 3093 |
This theorem is referenced by: alephreg 9802 arch 11704 harmonicbnd 25283 harmonicbnd2 25284 nbgrnself2 26845 heiborlem8 34544 fourierdlem62 41890 srhmsubclem1 43714 srhmsubc 43717 srhmsubcALTV 43735 |
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