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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 2372. (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 1918 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1918 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | cdeqnot.1 | . . . 4 ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | cdeqri 3760 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
5 | 1, 2, 4 | cbvab 2809 | . 2 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
6 | 5 | cdeqth 3761 | 1 ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 {cab 2710 CondEqwcdeq 3757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2372 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-cdeq 3758 |
This theorem is referenced by: (None) |
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