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Mirrors > Home > MPE Home > Th. List > iineq2d | Structured version Visualization version GIF version |
Description: Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.) |
Ref | Expression |
---|---|
iineq2d.1 | ⊢ Ⅎ𝑥𝜑 |
iineq2d.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
iineq2d | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iineq2d.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | iineq2d.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
3 | 2 | ex 416 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)) |
4 | 1, 3 | ralrimi 3180 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) |
5 | iineq2 4901 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) | |
6 | 4, 5 | syl 17 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 ∩ ciin 4882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-iin 4884 |
This theorem is referenced by: iineq2dv 4906 pmapglbx 37065 smflimmpt 43441 smfsupmpt 43446 smfinfmpt 43450 smflimsuplem4 43454 smflimsupmpt 43460 smfliminfmpt 43463 |
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