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Mirrors > Home > MPE Home > Th. List > sbcbi1 | Structured version Visualization version GIF version |
Description: Distribution of class substitution over biconditional. One direction of sbcbig 3820 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Ref | Expression |
---|---|
sbcbi1 | ⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3779 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → 𝐴 ∈ V) | |
2 | sbcbig 3820 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | |
3 | 2 | biimpd 230 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
4 | 1, 3 | mpcom 38 | 1 ⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 Vcvv 3492 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-v 3494 df-sbc 3770 |
This theorem is referenced by: dfconngr1 27894 |
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