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 2931 | . 2 ⊢ (𝑥 ∈ 𝐵 → 𝜑) |
4 | 1, 3 | vtoclga 3272 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∀wral 2912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-v 3202 |
This theorem is referenced by: alephreg 9404 arch 11289 harmonicbnd 24730 harmonicbnd2 24731 heiborlem8 33617 fourierdlem62 40385 srhmsubclem1 42073 srhmsubc 42076 srhmsubcALTV 42094 |
Copyright terms: Public domain | W3C validator |