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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcbiVD | Structured version Visualization version GIF version |
Description: Implication form of sbcbii 3829.
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 40893 is sbcbiVD 41230 without virtual deductions and was automatically
derived from sbcbiVD 41230.
|
Ref | Expression |
---|---|
sbcbiVD | ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 40928 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
2 | idn2 40967 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓) ▶ ∀𝑥(𝜑 ↔ 𝜓) ) | |
3 | spsbc 3785 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → [𝐴 / 𝑥](𝜑 ↔ 𝜓))) | |
4 | 1, 2, 3 | e12 41078 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓) ▶ [𝐴 / 𝑥](𝜑 ↔ 𝜓) ) |
5 | sbcbig 3823 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) | |
6 | 5 | biimpd 231 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
7 | 1, 4, 6 | e12 41078 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 , ∀𝑥(𝜑 ↔ 𝜓) ▶ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓) ) |
8 | 7 | in2 40959 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)) ) |
9 | 8 | in1 40925 | 1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∈ wcel 2114 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-sbc 3773 df-vd1 40924 df-vd2 40932 |
This theorem is referenced by: (None) |
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