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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 3437 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 3404 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | sbcie2g.2 | . 2 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) | |
3 | sbsbc 3406 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
4 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | sbcie2g.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | sbie 2396 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
7 | 3, 6 | bitr3i 265 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
8 | 1, 2, 7 | vtoclbg 3240 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒)) |
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-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: sbcel2gv 3463 csbie2g 3530 brab1 4630 bnj90 30042 bnj124 30195 riotasvd 33260 |
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