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Mirrors > Home > NFE Home > Th. List > cbvsbc | GIF version |
Description: Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
cbvsbc.1 | ⊢ Ⅎyφ |
cbvsbc.2 | ⊢ Ⅎxψ |
cbvsbc.3 | ⊢ (x = y → (φ ↔ ψ)) |
Ref | Expression |
---|---|
cbvsbc | ⊢ ([̣A / x]̣φ ↔ [̣A / y]̣ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvsbc.1 | . . . 4 ⊢ Ⅎyφ | |
2 | cbvsbc.2 | . . . 4 ⊢ Ⅎxψ | |
3 | cbvsbc.3 | . . . 4 ⊢ (x = y → (φ ↔ ψ)) | |
4 | 1, 2, 3 | cbvab 2471 | . . 3 ⊢ {x ∣ φ} = {y ∣ ψ} |
5 | 4 | eleq2i 2417 | . 2 ⊢ (A ∈ {x ∣ φ} ↔ A ∈ {y ∣ ψ}) |
6 | df-sbc 3047 | . 2 ⊢ ([̣A / x]̣φ ↔ A ∈ {x ∣ φ}) | |
7 | df-sbc 3047 | . 2 ⊢ ([̣A / y]̣ψ ↔ A ∈ {y ∣ ψ}) | |
8 | 5, 6, 7 | 3bitr4i 268 | 1 ⊢ ([̣A / x]̣φ ↔ [̣A / y]̣ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 Ⅎwnf 1544 ∈ wcel 1710 {cab 2339 [̣wsbc 3046 |
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-sbc 3047 |
This theorem is referenced by: cbvsbcv 3075 cbvcsb 3140 |
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