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Mirrors > Home > MPE Home > Th. List > Mathboxes > csbeq2gVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of csbeq2 3793.
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 3793 is csbeq2gVD 42034 without virtual deductions and was
automatically derived from csbeq2gVD 42034.
|
Ref | Expression |
---|---|
csbeq2gVD | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 41716 | . . . 4 ⊢ ( 𝐴 ∈ 𝑉 ▶ 𝐴 ∈ 𝑉 ) | |
2 | spsbc 3692 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → [𝐴 / 𝑥]𝐵 = 𝐶)) | |
3 | 1, 2 | e1a 41769 | . . 3 ⊢ ( 𝐴 ∈ 𝑉 ▶ (∀𝑥 𝐵 = 𝐶 → [𝐴 / 𝑥]𝐵 = 𝐶) ) |
4 | sbceqg 4296 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) | |
5 | 1, 4 | e1a 41769 | . . 3 ⊢ ( 𝐴 ∈ 𝑉 ▶ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) ) |
6 | imbi2 352 | . . . 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 252 | . . 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 41830 | . 2 ⊢ ( 𝐴 ∈ 𝑉 ▶ (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) ) |
9 | 8 | in1 41713 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1540 = wceq 1542 ∈ wcel 2113 [wsbc 3679 ⦋csb 3788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-sbc 3680 df-csb 3789 df-vd1 41712 |
This theorem is referenced by: (None) |
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