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Mirrors > Home > MPE Home > Th. List > Mathboxes > csbeq2gVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of csbeq2 3889.
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.
csbeq2 3889 is csbeq2gVD 44396 without virtual deductions and was
automatically derived from csbeq2gVD 44396.
|
Ref | Expression |
---|---|
csbeq2gVD | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 44078 | . . . 4 ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 ) | |
2 | spsbc 3781 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → [𝐴 / 𝑥]𝐵 = 𝐶)) | |
3 | 1, 2 | e1a 44131 | . . 3 ⊢ ( 𝐴 ∈ 𝑉 ▶ (∀𝑥 𝐵 = 𝐶 → [𝐴 / 𝑥]𝐵 = 𝐶) ) |
4 | sbceqg 4405 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) | |
5 | 1, 4 | e1a 44131 | . . 3 ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) ) |
6 | imbi2 347 | . . . 4 ⊢ (([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) → ((∀𝑥 𝐵 = 𝐶 → [𝐴 / 𝑥]𝐵 = 𝐶) ↔ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))) | |
7 | 6 | biimpcd 248 | . . 3 ⊢ ((∀𝑥 𝐵 = 𝐶 → [𝐴 / 𝑥]𝐵 = 𝐶) → (([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))) |
8 | 3, 5, 7 | e11 44192 | . 2 ⊢ ( 𝐴 ∈ 𝑉 ▶ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) ) |
9 | 8 | in1 44075 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 = wceq 1533 ∈ wcel 2098 [wsbc 3768 ⦋csb 3884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-sbc 3769 df-csb 3885 df-vd1 44074 |
This theorem is referenced by: (None) |
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