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Mirrors > Home > MPE Home > Th. List > sbcied2 | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Ref | Expression |
---|---|
sbcied2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sbcied2.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
sbcied2.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
sbcied2 | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcied2.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
3 | sbcied2.2 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 2, 3 | sylan9eqr 2795 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵) |
5 | sbcied2.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒)) | |
6 | 4, 5 | syldan 592 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
7 | 1, 6 | sbcied 3823 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 [wsbc 3778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-sbc 3779 |
This theorem is referenced by: iscat 17616 sectffval 17697 issubc 17785 isfunc 17814 cat1 18047 ismgm 18562 issgrp 18611 isnsg 19035 isring 20060 islbs 20687 isdomn 20910 isassa 21411 opsrval 21601 isuhgr 28320 isushgr 28321 isupgr 28344 isumgr 28355 isuspgr 28412 isusgr 28413 isrng 46650 isthinc 47641 |
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