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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnopabeqd | Structured version Visualization version GIF version |
Description: Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.) |
Ref | Expression |
---|---|
fnopabeqd.1 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
fnopabeqd | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnopabeqd.1 | . . . 4 ⊢ (𝜑 → 𝐵 = 𝐶) | |
2 | 1 | eqeq2d 2834 | . . 3 ⊢ (𝜑 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶)) |
3 | 2 | anbi2d 630 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))) |
4 | 3 | opabbidv 5134 | 1 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {copab 5130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-opab 5131 |
This theorem is referenced by: (None) |
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