Mathbox for Jonathan Ben-Naim |
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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 3497 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | bnj610.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓′)) | |
3 | 1, 2 | sbcie 3811 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓′) |
4 | 3 | sbcbii 3828 | . 2 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓′) |
5 | sbccow 3794 | . 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 3811 | . 2 ⊢ ([𝐴 / 𝑦]𝜓′ ↔ 𝜓) |
9 | 4, 5, 8 | 3bitr3i 303 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 Vcvv 3494 [wsbc 3771 |
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 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 df-sbc 3772 |
This theorem is referenced by: bnj611 32185 bnj1000 32208 |
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