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Mirrors > Home > ILE Home > Th. List > sbcbid | GIF version |
Description: Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |
Ref | Expression |
---|---|
sbcbid.1 | ⊢ Ⅎ𝑥𝜑 |
sbcbid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
sbcbid | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcbid.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | sbcbid.2 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | abbid 2195 | . . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
4 | 3 | eleq2d 2148 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ {𝑥 ∣ 𝜒})) |
5 | df-sbc 2816 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
6 | df-sbc 2816 | . 2 ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝐴 ∈ {𝑥 ∣ 𝜒}) | |
7 | 4, 5, 6 | 3bitr4g 221 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 Ⅎwnf 1389 ∈ wcel 1433 {cab 2067 [wsbc 2815 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-sbc 2816 |
This theorem is referenced by: sbcbidv 2872 csbeq2d 2930 bezoutlemstep 10386 |
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