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Mirrors > Home > MPE Home > Th. List > sbcie | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |
Ref | Expression |
---|---|
sbcie.1 | ⊢ 𝐴 ∈ V |
sbcie.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbcie | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcie.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sbcie.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | sbcieg 3435 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 [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: tfinds2 6955 findcard2 8085 ac6sfi 8089 ac6num 9184 fpwwe 9347 nn1suc 10918 wrdind 13328 cjth 13691 fprodser 14518 prmind2 15236 joinlem 16834 meetlem 16848 mrcmndind 17189 isghm 17483 islmod 18690 islindf 19970 fgcl 21492 cfinfil 21507 csdfil 21508 supfil 21509 fin1aufil 21546 quotval 23851 bnj62 30040 bnj610 30071 bnj976 30102 bnj106 30192 bnj125 30196 bnj154 30202 bnj155 30203 bnj526 30212 bnj540 30216 bnj591 30235 bnj609 30241 bnj893 30252 bnj1417 30363 soseq 30995 poimirlem27 32606 sdclem2 32708 fdc 32711 fdc1 32712 lshpkrlem3 33417 hdmap1fval 36104 hdmapfval 36137 rabren3dioph 36397 2nn0ind 36528 zindbi 36529 onfrALTlem5 37778 dfconngr1 41355 isconngr 41356 isconngr1 41357 |
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