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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 3831 | . 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 206 = wceq 1536 ∈ wcel 2105 Vcvv 3477 [wsbc 3790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-sbc 3791 |
This theorem is referenced by: sbc2ie 3873 csbie 3943 rexopabb 5537 reuop 6314 tfinds2 7884 soseq 8182 findcard2 9202 ac6sfi 9317 ac6num 10516 fpwwe 10683 nn1suc 12285 wrdind 14756 cjth 15138 fprodser 15981 prmind2 16718 joinlem 18440 meetlem 18454 mndind 18853 isghm 19245 isghmOLD 19246 islmod 20878 islindf 21849 fgcl 23901 cfinfil 23916 csdfil 23917 supfil 23918 fin1aufil 23955 quotval 26348 dfconngr1 30216 isconngr 30217 isconngr1 30218 wrdt2ind 32922 bnj62 34712 bnj610 34739 bnj976 34769 bnj106 34860 bnj125 34864 bnj154 34870 bnj155 34871 bnj526 34880 bnj540 34884 bnj591 34903 bnj609 34909 bnj893 34920 bnj1417 35033 poimirlem27 37633 sdclem2 37728 fdc 37731 fdc1 37732 lshpkrlem3 39093 hdmap1fval 41778 hdmapfval 41809 sn-isghm 42659 rabren3dioph 42802 2nn0ind 42933 zindbi 42934 onfrALTlem5 44539 onfrALTlem5VD 44882 reupr 47446 |
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