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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 3470 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 3437 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | sbcie2g.2 | . 2 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) | |
3 | sbsbc 3439 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
4 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | sbcie2g.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | sbie 2408 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
7 | 3, 6 | bitr3i 266 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
8 | 1, 2, 7 | vtoclbg 3267 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 [wsb 1880 ∈ wcel 1990 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-sbc 3436 |
This theorem is referenced by: sbcel2gv 3496 csbie2g 3564 brab1 4700 bnj90 30788 bnj124 30941 riotasvd 34242 |
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