Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3207 | . 2 ⊢ (𝑥 ∈ 𝐵 → 𝜑) |
4 | 1, 3 | vtoclga 3574 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2814 df-clel 2893 df-ral 3143 |
This theorem is referenced by: alephreg 10004 arch 11895 harmonicbnd 25581 harmonicbnd2 25582 nbgrnself2 27142 heiborlem8 35111 fourierdlem62 42473 srhmsubclem1 44364 srhmsubc 44367 srhmsubcALTV 44385 |
Copyright terms: Public domain | W3C validator |