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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 3080 | . 2 ⊢ (𝑥 ∈ 𝐵 → 𝜑) |
4 | 1, 3 | vtoclga 3423 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1631 ∈ wcel 2145 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-v 3353 |
This theorem is referenced by: alephreg 9604 arch 11489 harmonicbnd 24944 harmonicbnd2 24945 nbgrnself2 26472 heiborlem8 33942 fourierdlem62 40895 srhmsubclem1 42594 srhmsubc 42597 srhmsubcALTV 42615 |
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