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Mirrors > Home > MPE Home > Th. List > sbcbr2g | Structured version Visualization version GIF version |
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) |
Ref | Expression |
---|---|
sbcbr2g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ 𝐵𝑅⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcbr12g 5130 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵𝑅⦋𝐴 / 𝑥⦌𝐶)) | |
2 | csbconstg 3851 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) | |
3 | 2 | breq1d 5084 | . 2 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐵𝑅⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐵𝑅⦋𝐴 / 𝑥⦌𝐶)) |
4 | 1, 3 | bitrd 278 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ 𝐵𝑅⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 [wsbc 3716 ⦋csb 3832 class class class wbr 5074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 |
This theorem is referenced by: prmgaplem7 16758 telgsums 19594 fvmptnn04if 21998 bnj110 32838 frege124d 41369 frege72 41543 frege91 41562 frege116 41587 frege120 41591 |
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