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Mirrors > Home > MPE Home > Th. List > sbcie2g | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3815 avoids a disjointness condition on 𝑥, 𝐴 by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
sbcie2g.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
sbcie2g.2 | ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
sbcie2g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 3774 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | sbcie2g.2 | . 2 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) | |
3 | sbsbc 3776 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
4 | sbcie2g.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 4 | sbievw 2087 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 3, 5 | bitr3i 277 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
7 | 1, 2, 6 | vtoclbg 3539 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 [wsb 2059 ∈ wcel 2098 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-sbc 3773 |
This theorem is referenced by: sbcel2gv 3844 csbie2g 3931 brab1 5189 bnj90 34261 bnj124 34410 riotasvd 38338 |
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