![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sbcco2 | Structured version Visualization version GIF version |
Description: A composition law for class substitution. Importantly, 𝑥 may occur free in the class expression substituted for 𝐴. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
sbcco2.1 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
sbcco2 | ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbsbc 3780 | . 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝑥 / 𝑦][𝐵 / 𝑥]𝜑) | |
2 | sbcco2.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
3 | 2 | equcoms 2016 | . . . 4 ⊢ (𝑦 = 𝑥 → 𝐴 = 𝐵) |
4 | dfsbcq 3778 | . . . . 5 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑)) | |
5 | 4 | bicomd 222 | . . . 4 ⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝑦 = 𝑥 → ([𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
7 | 6 | sbievw 2088 | . 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) |
8 | 1, 7 | bitr3i 276 | 1 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 [wsb 2060 [wsbc 3776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-sbc 3777 |
This theorem is referenced by: tfinds2 7874 |
Copyright terms: Public domain | W3C validator |