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Mirrors > Home > MPE Home > Th. List > Mathboxes > csbeq2gOLD | Structured version Visualization version GIF version |
Description: Formula-building implication rule for class substitution. Closed form of csbeq2i 3945. csbeq2gOLD 37786 is derived from the virtual deduction proof csbeq2gVD 38150. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete version of csbeq2 3503 as of 11-Oct-2018. (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
csbeq2gOLD | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbc 3415 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → [𝐴 / 𝑥]𝐵 = 𝐶)) | |
2 | sbceqg 3936 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) | |
3 | 1, 2 | sylibd 228 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 = wceq 1475 ∈ wcel 1977 [wsbc 3402 ⦋csb 3499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sbc 3403 df-csb 3500 |
This theorem is referenced by: csbsngVD 38151 csbxpgVD 38152 csbresgVD 38153 csbrngVD 38154 csbima12gALTVD 38155 |
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