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Mirrors > Home > MPE Home > Th. List > sbceqal | Structured version Visualization version GIF version |
Description: Class version of one implication of equvelv 2015. (Contributed by Andrew Salmon, 28-Jun-2011.) |
Ref | Expression |
---|---|
sbceqal | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbc 3719 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵))) | |
2 | sbcimg 3749 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ ([𝐴 / 𝑥]𝑥 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵))) | |
3 | eqid 2795 | . . . . 5 ⊢ 𝐴 = 𝐴 | |
4 | eqsbc3 3746 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 = 𝐴)) | |
5 | 3, 4 | mpbiri 259 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝑥 = 𝐴) |
6 | pm5.5 363 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐴 → (([𝐴 / 𝑥]𝑥 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) ↔ [𝐴 / 𝑥]𝑥 = 𝐵)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑥 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) ↔ [𝐴 / 𝑥]𝑥 = 𝐵)) |
8 | eqsbc3 3746 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
9 | 2, 7, 8 | 3bitrd 306 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝑥 = 𝐵) ↔ 𝐴 = 𝐵)) |
10 | 1, 9 | sylibd 240 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1520 = wceq 1522 ∈ wcel 2081 [wsbc 3706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-sbc 3707 |
This theorem is referenced by: sbeqalb 3764 |
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