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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 3680 | . 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝑥 / 𝑦][𝐵 / 𝑥]𝜑) | |
2 | sbcco2.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
3 | 2 | equcoms 1978 | . . . 4 ⊢ (𝑦 = 𝑥 → 𝐴 = 𝐵) |
4 | dfsbcq 3678 | . . . . 5 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑)) | |
5 | 4 | bicomd 215 | . . . 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 2042 | . 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) |
8 | 1, 7 | bitr3i 269 | 1 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1508 [wsb 2016 [wsbc 3676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1744 df-sb 2017 df-clab 2754 df-cleq 2766 df-clel 2841 df-sbc 3677 |
This theorem is referenced by: tfinds2 7393 |
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