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Mirrors > Home > ILE Home > Th. List > sbcbii | GIF version |
Description: Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005.) |
Ref | Expression |
---|---|
sbcbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
sbcbii | ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓)) |
3 | 2 | sbcbidv 2967 | . 2 ⊢ (⊤ → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) |
4 | 3 | mptru 1340 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ⊤wtru 1332 [wsbc 2909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-sbc 2910 |
This theorem is referenced by: eqsbc3r 2969 sbc3an 2970 sbccomlem 2983 sbccom 2984 sbcabel 2990 csbco 3013 sbcnel12g 3019 sbcne12g 3020 sbccsbg 3031 sbccsb2g 3032 csbnestgf 3052 csbabg 3061 sbcssg 3472 sbcrel 4625 difopab 4672 sbcfung 5147 f1od2 6132 mpoxopovel 6138 bezoutlemnewy 11684 bezoutlemstep 11685 bezoutlemmain 11686 |
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