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Mirrors > Home > MPE Home > Th. List > sbcieg | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |
Ref | Expression |
---|---|
sbcieg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbcieg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | sbcieg.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | sbciegf 3434 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ 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-3an 1033 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: sbcie 3437 ralsng 4165 rexsng 4166 rabsnif 4202 ralrnmpt 6276 fpwwe2lem3 9334 nn1suc 10918 mrcmndind 17189 fgcl 21492 cfinfil 21507 csdfil 21508 supfil 21509 fin1aufil 21546 ifeqeqx 28745 nn0min 28954 bnj1452 30374 cdlemk35s 35243 cdlemk39s 35245 cdlemk42 35247 2nn0ind 36528 zindbi 36529 trsbcVD 38135 onfrALTlem5VD 38143 |
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