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Mirrors > Home > NFE Home > Th. List > vtoclgaf | GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Ref | Expression |
---|---|
vtoclgaf.1 | ⊢ ℲxA |
vtoclgaf.2 | ⊢ Ⅎxψ |
vtoclgaf.3 | ⊢ (x = A → (φ ↔ ψ)) |
vtoclgaf.4 | ⊢ (x ∈ B → φ) |
Ref | Expression |
---|---|
vtoclgaf | ⊢ (A ∈ B → ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtoclgaf.1 | . . 3 ⊢ ℲxA | |
2 | 1 | nfel1 2499 | . . . 4 ⊢ Ⅎx A ∈ B |
3 | vtoclgaf.2 | . . . 4 ⊢ Ⅎxψ | |
4 | 2, 3 | nfim 1813 | . . 3 ⊢ Ⅎx(A ∈ B → ψ) |
5 | eleq1 2413 | . . . 4 ⊢ (x = A → (x ∈ B ↔ A ∈ B)) | |
6 | vtoclgaf.3 | . . . 4 ⊢ (x = A → (φ ↔ ψ)) | |
7 | 5, 6 | imbi12d 311 | . . 3 ⊢ (x = A → ((x ∈ B → φ) ↔ (A ∈ B → ψ))) |
8 | vtoclgaf.4 | . . 3 ⊢ (x ∈ B → φ) | |
9 | 1, 4, 7, 8 | vtoclgf 2913 | . 2 ⊢ (A ∈ B → (A ∈ B → ψ)) |
10 | 9 | pm2.43i 43 | 1 ⊢ (A ∈ B → ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 Ⅎwnf 1544 = wceq 1642 ∈ wcel 1710 Ⅎwnfc 2476 |
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-v 2861 |
This theorem is referenced by: vtoclga 2920 ssiun2s 4010 fvmptss 5705 fvmptf 5722 |
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