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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.) |
Ref | Expression |
---|---|
sbc5 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3782 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
2 | exsimpl 1865 | . . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃𝑥 𝑥 = 𝐴) | |
3 | isset 3507 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
4 | 2, 3 | sylibr 236 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝐴 ∈ V) |
5 | dfsbcq2 3775 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
6 | eqeq2 2833 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
7 | 6 | anbi1d 631 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑))) |
8 | 7 | exbidv 1918 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
9 | sb5 2272 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
10 | 5, 8, 9 | vtoclbg 3569 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
11 | 1, 4, 10 | pm5.21nii 382 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 [wsb 2065 ∈ wcel 2110 Vcvv 3495 [wsbc 3772 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3497 df-sbc 3773 |
This theorem is referenced by: sbc6g 3801 sbc7 3803 sbciegft 3808 sbccomlem 3853 csb2 3885 rexsns 4603 sbcop1 5372 sbccom2lem 35396 pm13.192 40735 pm13.195 40738 2sbc5g 40741 iotasbc 40744 pm14.122b 40748 iotasbc5 40756 sbcpr 43676 |
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