Theorem abbi1 2801

Description: Equivalent formulas yield equal class abstractions (closed form). This is the forward implication of abbi 2805, proved from fewer axioms. (Contributed by BJ and WL and SN, 20-Aug-2023.)

Assertion

Ref	Expression
abbi1	⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})

Proof of Theorem abbi1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		spsbbi 2082	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
2		df-clab 2717	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
3		df-clab 2717	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
4	1, 2, 3	3bitr4g 317	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜓}))
5	4	eqrdv 2736	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1540   = wceq 1542  [wsb 2073   ∈ wcel 2113  {cab 2716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-9 2123  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730

This theorem is referenced by:  abbidv  2802  abbii  2803  abbid  2804  sbcbi2  3741  iuneq12df  4908  iotabi  6312  uniabio  6313  iotanul  6318  iuneq12daf  30470  bj-abv  34720  bj-cleq  34772  iotain  41565