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Mirrors > Home > NFE Home > Th. List > hblem | GIF version |
Description: Change the free variable of a hypothesis builder. Lemma for nfcrii 2482. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
hblem.1 | ⊢ (y ∈ A → ∀x y ∈ A) |
Ref | Expression |
---|---|
hblem | ⊢ (z ∈ A → ∀x z ∈ A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hblem.1 | . . 3 ⊢ (y ∈ A → ∀x y ∈ A) | |
2 | 1 | hbsb 2110 | . 2 ⊢ ([z / y]y ∈ A → ∀x[z / y]y ∈ A) |
3 | clelsb3 2455 | . 2 ⊢ ([z / y]y ∈ A ↔ z ∈ A) | |
4 | 3 | albii 1566 | . 2 ⊢ (∀x[z / y]y ∈ A ↔ ∀x z ∈ A) |
5 | 2, 3, 4 | 3imtr3i 256 | 1 ⊢ (z ∈ A → ∀x z ∈ A) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 [wsb 1648 ∈ 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-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-cleq 2346 df-clel 2349 |
This theorem is referenced by: nfcrii 2482 |
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