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Mirrors > Home > MPE Home > Th. List > eqsbc3 | Structured version Visualization version GIF version |
Description: Substitution applied to an atomic wff. Class version of eqsb3 2936. (Contributed by Andrew Salmon, 29-Jun-2011.) Avoid ax-13 2381. (Revised by Wolf Lammen, 29-Apr-2023.) |
Ref | Expression |
---|---|
eqsbc3 | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 3771 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ [𝐴 / 𝑥]𝑥 = 𝐵)) | |
2 | eqeq1 2822 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵)) | |
3 | sbsbc 3773 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ [𝑦 / 𝑥]𝑥 = 𝐵) | |
4 | eqsb3 2936 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ 𝑦 = 𝐵) | |
5 | 3, 4 | bitr3i 278 | . 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ 𝑦 = 𝐵) |
6 | 1, 2, 5 | vtoclbg 3566 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 [wsb 2060 ∈ wcel 2105 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-sbc 3770 |
This theorem is referenced by: sbceqal 3832 eqsbc3r 3834 fmptsnd 6923 fvmptnn04if 21385 snfil 22400 f1omptsnlem 34499 mptsnunlem 34501 topdifinffinlem 34510 relowlpssretop 34527 iotavalb 40639 onfrALTlem5 40753 eqsbc3rVD 41051 onfrALTlem5VD 41096 |
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