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Mirrors > Home > MPE Home > Th. List > sbciedOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sbcied 3760 as of 12-Oct-2024. (Contributed by NM, 13-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbciedOLD.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sbciedOLD.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
sbciedOLD | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbciedOLD.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | sbciedOLD.2 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | nfv 1917 | . 2 ⊢ Ⅎ𝑥𝜑 | |
4 | nfvd 1918 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜒) | |
5 | 1, 2, 3, 4 | sbciedf 3759 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 [wsbc 3715 |
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-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-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-sbc 3716 |
This theorem is referenced by: (None) |
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