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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 3775 | . 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝑥 / 𝑦][𝐵 / 𝑥]𝜑) | |
2 | sbcco2.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
3 | 2 | equcoms 2023 | . . . 4 ⊢ (𝑦 = 𝑥 → 𝐴 = 𝐵) |
4 | dfsbcq 3773 | . . . . 5 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑)) | |
5 | 4 | bicomd 225 | . . . 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 2099 | . 2 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) |
8 | 1, 7 | bitr3i 279 | 1 ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 [wsb 2065 [wsbc 3771 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-sbc 3772 |
This theorem is referenced by: tfinds2 7572 |
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