Theorem abbi 2801

Description: Equivalent formulas yield equal class abstractions (closed form). This is the backward implication of abbib 2805, 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 2079	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
2		df-clab 2715	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
3		df-clab 2715	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
4	1, 2, 3	3bitr4g 314	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝑦 ∈ {𝑥 ∣ 𝜓}))
5	4	eqrdv 2734	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542  [wsb 2068   ∈ wcel 2114  {cab 2714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728

This theorem is referenced by:  abbidv  2802  abbii  2803  abbid  2804  eqab  2874  sbcbi2  3787  iuneq12df  4960  axrep6g  5225  iotabi  6467  uniabio  6468  iotaval  6472  iotanul  6478  iuneq12daf  32626  bj-abv  37213  bj-cleq  37269  abbi1sn  42664  iotain  44844