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Mirrors > Home > MPE Home > Th. List > eqsbc1 | Structured version Visualization version GIF version |
Description: Substitution for the left-hand side in an equality. Class version of eqsb1 2865. (Contributed by Andrew Salmon, 29-Jun-2011.) Avoid ax-13 2375. (Revised by Wolf Lammen, 29-Apr-2023.) |
Ref | Expression |
---|---|
eqsbc1 | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 3793 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ [𝐴 / 𝑥]𝑥 = 𝐵)) | |
2 | eqeq1 2739 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵)) | |
3 | sbsbc 3795 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ [𝑦 / 𝑥]𝑥 = 𝐵) | |
4 | eqsb1 2865 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ 𝑦 = 𝐵) | |
5 | 3, 4 | bitr3i 277 | . 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ 𝑦 = 𝐵) |
6 | 1, 2, 5 | vtoclbg 3557 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 [wsb 2062 ∈ wcel 2106 [wsbc 3791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-sbc 3792 |
This theorem is referenced by: sbceqalOLD 3858 eqsbc2 3860 fmptsnd 7189 fvmptnn04if 22871 snfil 23888 f1omptsnlem 37319 mptsnunlem 37321 topdifinffinlem 37330 relowlpssretop 37347 iotavalb 44426 onfrALTlem5 44540 eqsbc2VD 44838 onfrALTlem5VD 44883 |
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