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Mirrors > Home > MPE Home > Th. List > eqsbc3 | Structured version Visualization version GIF version |
Description: Substitution applied to an atomic wff. Set theory version of eqsb3 2715. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
eqsbc3 | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 3404 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ [𝐴 / 𝑥]𝑥 = 𝐵)) | |
2 | eqeq1 2614 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵)) | |
3 | sbsbc 3406 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ [𝑦 / 𝑥]𝑥 = 𝐵) | |
4 | eqsb3 2715 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ 𝑦 = 𝐵) | |
5 | 3, 4 | bitr3i 265 | . 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐵 ↔ 𝑦 = 𝐵) |
6 | 1, 2, 5 | vtoclbg 3240 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 [wsb 1867 ∈ wcel 1977 [wsbc 3402 |
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-v 3175 df-sbc 3403 |
This theorem is referenced by: sbceqal 3454 eqsbc3r 3459 eqsbc3rOLD 3460 fmptsnd 6340 fvmptnn04if 20473 snfil 21478 f1omptsnlem 32359 mptsnunlem 32361 topdifinffinlem 32371 relowlpssretop 32388 iotavalb 37653 onfrALTlem5 37778 eqsbc3rVD 38097 onfrALTlem5VD 38143 |
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