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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj610 | Structured version Visualization version GIF version |
Description: Pass from equality (𝑥 = 𝐴) to substitution ([𝐴 / 𝑥]) without the distinct variable restriction ($d 𝐴 𝑥). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj610.1 | ⊢ 𝐴 ∈ V |
bnj610.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
bnj610.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓′)) |
bnj610.4 | ⊢ (𝑦 = 𝐴 → (𝜓′ ↔ 𝜓)) |
Ref | Expression |
---|---|
bnj610 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3176 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | bnj610.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓′)) | |
3 | 1, 2 | sbcie 3437 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓′) |
4 | 3 | sbcbii 3458 | . 2 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓′) |
5 | sbcco 3425 | . 2 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) | |
6 | bnj610.1 | . . 3 ⊢ 𝐴 ∈ V | |
7 | bnj610.4 | . . 3 ⊢ (𝑦 = 𝐴 → (𝜓′ ↔ 𝜓)) | |
8 | 6, 7 | sbcie 3437 | . 2 ⊢ ([𝐴 / 𝑦]𝜓′ ↔ 𝜓) |
9 | 4, 5, 8 | 3bitr3i 289 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 [wsbc 3402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 df-sbc 3403 |
This theorem is referenced by: bnj611 30242 bnj1000 30265 |
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