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Mirrors > Home > NFE Home > Th. List > vtoclegft | GIF version |
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2927.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) |
Ref | Expression |
---|---|
vtoclegft | ⊢ ((A ∈ B ∧ Ⅎxφ ∧ ∀x(x = A → φ)) → φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2869 | . . . 4 ⊢ (A ∈ B → ∃x x = A) | |
2 | exim 1575 | . . . 4 ⊢ (∀x(x = A → φ) → (∃x x = A → ∃xφ)) | |
3 | 1, 2 | mpan9 455 | . . 3 ⊢ ((A ∈ B ∧ ∀x(x = A → φ)) → ∃xφ) |
4 | 3 | 3adant2 974 | . 2 ⊢ ((A ∈ B ∧ Ⅎxφ ∧ ∀x(x = A → φ)) → ∃xφ) |
5 | 19.9t 1779 | . . 3 ⊢ (Ⅎxφ → (∃xφ ↔ φ)) | |
6 | 5 | 3ad2ant2 977 | . 2 ⊢ ((A ∈ B ∧ Ⅎxφ ∧ ∀x(x = A → φ)) → (∃xφ ↔ φ)) |
7 | 4, 6 | mpbid 201 | 1 ⊢ ((A ∈ B ∧ Ⅎxφ ∧ ∀x(x = A → φ)) → φ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ w3a 934 ∀wal 1540 ∃wex 1541 Ⅎwnf 1544 = wceq 1642 ∈ wcel 1710 |
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-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-3an 936 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-v 2861 |
This theorem is referenced by: (None) |
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