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Mirrors > Home > MPE Home > Th. List > csbiegf | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
csbiegf.1 | ⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶) |
csbiegf.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbiegf | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbiegf.2 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
2 | 1 | ax-gen 1796 | . 2 ⊢ ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) |
3 | csbiegf.1 | . . 3 ⊢ (𝐴 ∈ 𝑉 → Ⅎ𝑥𝐶) | |
4 | csbiebt 3923 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) | |
5 | 3, 4 | mpdan 684 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
6 | 2, 5 | mpbii 232 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1538 = wceq 1540 ∈ wcel 2105 Ⅎwnfc 2882 ⦋csb 3893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-v 3475 df-sbc 3778 df-csb 3894 |
This theorem is referenced by: csbief 3928 sbcco3gw 4422 sbcco3g 4427 csbco3g 4428 fmptcof 7130 fmpoco 8086 sumsnf 15696 prodsn 15913 prodsnf 15915 bpolylem 15999 pcmpt 16832 chfacfpmmulfsupp 22684 elmptrab 23650 dvfsumrlim3 25887 itgsubstlem 25902 itgsubst 25903 ifeqeqx 32206 disjunsn 32257 sbcaltop 35422 unirep 37045 cdleme31so 39713 cdleme31sn 39714 cdleme31sn1 39715 cdleme31se 39716 cdleme31se2 39717 cdleme31sc 39718 cdleme31sde 39719 cdleme31sn2 39723 cdlemeg47rv2 39844 cdlemk41 40254 monotuz 42142 oddcomabszz 42145 |
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