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Mirrors > Home > MPE Home > Th. List > sbcgf | Structured version Visualization version GIF version |
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
sbcgf.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
sbcgf | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcgf.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | sbctt 3467 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | |
3 | 1, 2 | mpan2 703 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 Ⅎwnf 1699 ∈ wcel 1977 [wsbc 3402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 df-sbc 3403 |
This theorem is referenced by: sbc19.21g 3469 sbcg 3470 sbcabel 3483 bnj110 30182 bnj1039 30293 sbali 33085 sbexi 33086 sbcgfi 33103 |
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