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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 3792 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
2 | clelab 2885 | . 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 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 [wsbc 3791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-sbc 3792 |
This theorem is referenced by: sbc6gOLD 3822 sbc7 3824 sbciegftOLD 3830 sbccomlemOLD 3879 csb2 3910 rexsns 4676 sbcop1 5499 sbccom2lem 38111 pm13.192 44406 pm13.195 44409 2sbc5g 44412 iotasbc 44415 pm14.122b 44419 iotasbc5 44427 sbcpr 47446 |
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