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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 3828 | . 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 1540 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 |
| This theorem is referenced by: sbc2ie 3866 csbie 3934 rexopabb 5533 reuop 6313 tfinds2 7885 soseq 8184 findcard2 9204 ac6sfi 9320 ac6num 10519 fpwwe 10686 nn1suc 12288 wrdind 14760 cjth 15142 fprodser 15985 prmind2 16722 joinlem 18428 meetlem 18442 mndind 18841 isghm 19233 isghmOLD 19234 islmod 20862 islindf 21832 fgcl 23886 cfinfil 23901 csdfil 23902 supfil 23903 fin1aufil 23940 quotval 26334 dfconngr1 30207 isconngr 30208 isconngr1 30209 wrdt2ind 32938 bnj62 34734 bnj610 34761 bnj976 34791 bnj106 34882 bnj125 34886 bnj154 34892 bnj155 34893 bnj526 34902 bnj540 34906 bnj591 34925 bnj609 34931 bnj893 34942 bnj1417 35055 poimirlem27 37654 sdclem2 37749 fdc 37752 fdc1 37753 lshpkrlem3 39113 hdmap1fval 41798 hdmapfval 41829 sn-isghm 42683 rabren3dioph 42826 2nn0ind 42957 zindbi 42958 onfrALTlem5 44562 onfrALTlem5VD 44905 reupr 47509 |
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