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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 3753 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) | |
3 | 1, 2 | mpan2 691 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 Ⅎwnf 1790 ∈ wcel 2113 [wsbc 3682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-sbc 3683 |
This theorem is referenced by: sbc19.21g 3755 sbcg 3756 sbcgfi 3757 sbcabel 3770 2nreu 4332 |
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