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Mirrors > Home > MPE Home > Th. List > cdeqab1 | Structured version Visualization version GIF version |
Description: Distribute conditional equality over abstraction. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdeqnot.1 | ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cdeqab1 | ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1916 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1916 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | cdeqnot.1 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | cdeqri 3712 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
5 | 1, 2, 4 | cbvab 2812 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
6 | 5 | cdeqth 3713 | 1 ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1540 {cab 2713 CondEqwcdeq 3709 |
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 1912 ax-6 1970 ax-7 2010 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-13 2370 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2714 df-cleq 2728 df-cdeq 3710 |
This theorem is referenced by: (None) |
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