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Mirrors > Home > MPE Home > Th. List > cdeqal1 | Structured version Visualization version GIF version |
Description: Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
cdeqnot.1 | ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cdeqal1 | ⊢ CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdeqnot.1 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | cdeqri 3666 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
3 | 2 | cbvalv 2331 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
4 | 3 | cdeqth 3667 | 1 ⊢ CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∀wal 1505 CondEqwcdeq 3663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-11 2093 ax-12 2106 ax-13 2301 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-nf 1747 df-cdeq 3664 |
This theorem is referenced by: (None) |
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