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Mirrors > Home > MPE Home > Th. List > Mathboxes > rabeq12f | Structured version Visualization version GIF version |
Description: Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.) |
Ref | Expression |
---|---|
rabeq12f.1 | ⊢ Ⅎ𝑥𝐴 |
rabeq12f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
rabeq12f | ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbi 3295 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}) | |
2 | 1 | biimpi 219 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}) |
3 | rabeq12f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
4 | rabeq12f.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
5 | 3, 4 | rabeqf 3391 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓}) |
6 | 2, 5 | sylan9eqr 2800 | 1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 Ⅎwnfc 2884 ∀wral 3061 {crab 3065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rab 3070 |
This theorem is referenced by: (None) |
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