|   | Mathbox for Alan Sare | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcbiVD | Structured version Visualization version GIF version | ||
| Description: Implication form of sbcbii 3846.
     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 44559 is sbcbiVD 44896 without virtual deductions and was automatically
     derived from sbcbiVD 44896. 
 | 
| Ref | Expression | 
|---|---|
| sbcbiVD | ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idn1 44594 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
| 2 | idn2 44633 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓) ▶ ∀𝑥(𝜑 ↔ 𝜓) ) | |
| 3 | spsbc 3801 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → [𝐴 / 𝑥](𝜑 ↔ 𝜓))) | |
| 4 | 1, 2, 3 | e12 44744 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓) ▶ [𝐴 / 𝑥](𝜑 ↔ 𝜓) ) | 
| 5 | sbcbig 3840 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | |
| 6 | 5 | biimpd 229 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | 
| 7 | 1, 4, 6 | e12 44744 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓) ▶ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) ) | 
| 8 | 7 | in2 44625 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) ) | 
| 9 | 8 | in1 44591 | 1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∈ wcel 2108 [wsbc 3788 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 df-vd1 44590 df-vd2 44598 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |