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Mirrors > Home > MPE Home > Th. List > csbconstgf | Structured version Visualization version GIF version |
Description: Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by NM, 10-Nov-2005.) |
Ref | Expression |
---|---|
csbconstgf.1 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
csbconstgf | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbconstgf.1 | . 2 ⊢ Ⅎ𝑥𝐵 | |
2 | csbtt 3903 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐵) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) | |
3 | 1, 2 | mpan2 689 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Ⅎwnfc 2964 ⦋csb 3886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-sbc 3776 df-csb 3887 |
This theorem is referenced by: csbconstg 3905 poimirlem25 34921 |
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