![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sbciegf | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
sbciegf.1 | ⊢ Ⅎ𝑥𝜓 |
sbciegf.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbciegf | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbciegf.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | sbciegf.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ax-gen 1839 | . 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
4 | sbciegft 3682 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | |
5 | 1, 3, 4 | mp3an23 1526 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1599 = wceq 1601 Ⅎwnf 1827 ∈ wcel 2106 [wsbc 3651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-12 2162 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-v 3399 df-sbc 3652 |
This theorem is referenced by: sbcieg 3684 rexsngf 4440 ralsngf 4441 opelopabgf 5232 opelopabf 5237 eqerlem 8060 bnj919 31436 bnj1464 31513 bnj1123 31653 bnj1373 31697 poimirlem25 34044 sbccomieg 38299 aomclem6 38570 fveqsb 39593 |
Copyright terms: Public domain | W3C validator |