Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cdeqel | Structured version Visualization version GIF version |
Description: Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
cdeqeq.1 | ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) |
cdeqeq.2 | ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
cdeqel | ⊢ CondEq(𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdeqeq.1 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵) | |
2 | 1 | cdeqri 3421 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
3 | cdeqeq.2 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷) | |
4 | 3 | cdeqri 3421 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
5 | 2, 4 | eleq12d 2695 | . 2 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
6 | 5 | cdeqi 3420 | 1 ⊢ CondEq(𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 CondEqwcdeq 3418 |
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-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-cleq 2615 df-clel 2618 df-cdeq 3419 |
This theorem is referenced by: nfccdeq 3433 |
Copyright terms: Public domain | W3C validator |