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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 2151 | . . 3 ⊢ (φ → {x ∣ ψ} = {x ∣ χ}) |
4 | 3 | eleq2d 2104 | . 2 ⊢ (φ → (A ∈ {x ∣ ψ} ↔ A ∈ {x ∣ χ})) |
5 | df-sbc 2759 | . 2 ⊢ ([A / x]ψ ↔ A ∈ {x ∣ ψ}) | |
6 | df-sbc 2759 | . 2 ⊢ ([A / x]χ ↔ A ∈ {x ∣ χ}) | |
7 | 4, 5, 6 | 3bitr4g 212 | 1 ⊢ (φ → ([A / x]ψ ↔ [A / x]χ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 Ⅎwnf 1346 ∈ wcel 1390 {cab 2023 [wsbc 2758 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-sbc 2759 |
This theorem is referenced by: sbcbidv 2811 csbeq2d 2868 |
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