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| Mirrors > Home > MPE Home > Th. List > sbc5ALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of sbc5 3816. This proof helps show how clelab 2887 works, since it is equivalent but shorter thanks to now-available library theorems like vtoclbg 3557 and isset 3494. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| sbc5ALT | ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbcex 3798 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
| 2 | exsimpl 1868 | . . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃𝑥 𝑥 = 𝐴) | |
| 3 | isset 3494 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
| 4 | 2, 3 | sylibr 234 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝐴 ∈ V) |
| 5 | dfsbcq2 3791 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
| 6 | eqeq2 2749 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
| 7 | 6 | anbi1d 631 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑))) |
| 8 | 7 | exbidv 1921 | . . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
| 9 | sb5 2276 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
| 10 | 5, 8, 9 | vtoclbg 3557 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) |
| 11 | 1, 4, 10 | pm5.21nii 378 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 [wsb 2064 ∈ wcel 2108 Vcvv 3480 [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-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sbc 3789 |
| This theorem is referenced by: (None) |
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