Theorem abbi 2827

Description: Equivalent formulas yield equal class abstractions (closed form). This is the backward implication of abbib 2831, proved from fewer axioms, and hence is independently named. (Contributed by BJ and WL and SN, 20-Aug-2023.)

Assertion

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

Proof of Theorem abbi

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		spsbbi 2106	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
2		df-clab 2741	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
3		df-clab 2741	. . 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 2760	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1558   = wceq 1560  [wsb 2090   ∈ wcel 2142  {cab 2740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754

This theorem is referenced by:  abbidv  2828  abbii  2829  abbid  2830  eqab  2900  sbcbi2  3802  iuneq12df  4976  axrep6g  5240  iotabi  6490  uniabio  6491  iotaval  6495  iotanul  6501  iuneq12daf  32756  bj-abv  37391  bj-cleq  37447  abbi1sn  42842  iotain  44993