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| Mirrors > Home > MPE Home > Th. List > sbc5 | Structured version Visualization version GIF version | ||
| Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by SN, 2-Sep-2024.) |
| Ref | Expression |
|---|---|
| sbc5 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-sbc 3789 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
| 2 | clelab 2887 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | |
| 3 | 1, 2 | bitri 275 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 [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-10 2141 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 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: sbc7 3820 sbciegftOLD 3826 sbccomlemOLD 3870 csb2 3901 rexsns 4671 sbcop1 5493 sbccom2lem 38131 pm13.192 44429 pm13.195 44432 2sbc5g 44435 iotasbc 44438 pm14.122b 44442 iotasbc5 44450 sbcpr 47508 |
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