Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
||
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 2632 | . . 3 ⊢ (𝜑 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶)) |
3 | 2 | anbi2d 740 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶))) |
4 | 3 | opabbidv 4716 | 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 384 = wceq 1483 ∈ wcel 1990 {copab 4712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-opab 4713 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |