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| 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 1795 | . 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| 4 | sbciegft 3825 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | |
| 5 | 1, 3, 4 | mp3an23 1455 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 [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-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 |
| This theorem is referenced by: rexsngf 4672 ralsngf 4673 opelopabgf 5545 opelopabf 5550 eqerlem 8780 bnj919 34781 bnj1464 34858 bnj1123 35000 bnj1373 35044 poimirlem25 37652 sbccomieg 42804 aomclem6 43071 fveqsb 44472 |
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