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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcbiVD | Structured version Visualization version GIF version |
Description: Implication form of sbcbii 3687.
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant.
sbcbi 39513 is sbcbiVD 39860 without virtual deductions and was automatically
derived from sbcbiVD 39860.
|
Ref | Expression |
---|---|
sbcbiVD | ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 39548 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
2 | idn2 39596 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓) ▶ ∀𝑥(𝜑 ↔ 𝜓) ) | |
3 | spsbc 3644 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → [𝐴 / 𝑥](𝜑 ↔ 𝜓))) | |
4 | 1, 2, 3 | e12 39708 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓) ▶ [𝐴 / 𝑥](𝜑 ↔ 𝜓) ) |
5 | sbcbig 3676 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | |
6 | 5 | biimpd 221 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
7 | 1, 4, 6 | e12 39708 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓) ▶ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) ) |
8 | 7 | in2 39588 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) ) |
9 | 8 | in1 39545 | 1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1651 ∈ wcel 2157 [wsbc 3631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-12 2213 ax-13 2354 ax-ext 2775 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-v 3385 df-sbc 3632 df-vd1 39544 df-vd2 39552 |
This theorem is referenced by: (None) |
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