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Mirrors > Home > MPE Home > Th. List > abbi1 | Structured version Visualization version GIF version |
Description: Equivalent formulas yield equal class abstractions (closed form). This is the forward implication of abbi 2887, proved from fewer axioms. (Contributed by BJ and WL and SN, 20-Aug-2023.) |
Ref | Expression |
---|---|
abbi1 | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spsbbi 2077 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) | |
2 | df-clab 2799 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
3 | df-clab 2799 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
4 | 1, 2, 3 | 3bitr4g 316 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜓})) |
5 | 4 | eqrdv 2818 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1534 = wceq 1536 [wsb 2068 ∈ wcel 2113 {cab 2798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-9 2123 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-sb 2069 df-clab 2799 df-cleq 2813 |
This theorem is referenced by: abbidv 2884 abbii 2885 abbid 2886 iuneq12df 4938 iotabi 6320 uniabio 6321 iotanul 6326 iuneq12daf 30308 bj-cleq 34298 iotain 40823 |
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