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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcbi | Structured version Visualization version GIF version |
Description: Implication form of sbcbii 3837. sbcbi 43290 is sbcbiVD 43627 without virtual deductions and was automatically derived from sbcbiVD 43627 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbcbi | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbc 3790 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝜓) → [𝐴 / 𝑥](𝜑 ↔ 𝜓))) | |
2 | sbcbig 3831 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | |
3 | 1, 2 | sylibd 238 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 ∈ wcel 2106 [wsbc 3777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-sbc 3778 |
This theorem is referenced by: trsbcVD 43628 sbcssgVD 43634 |
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