Theorem abbi 2825

Description: Equivalent formulas yield equal class abstractions (closed form). This is the backward implication of abbib 2829, 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 2110	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓))
2		df-clab 2739	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
3		df-clab 2739	. . 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 2758	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  [wsb 2099   ∈ wcel 2145  {cab 2738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752

This theorem is used by:  abbidv  2826  abbii  2827  abbid  2828  eqab  2898  sbcbi2  3797  iuneq12df  4978  axrep6g  5245  iotabi  6502  uniabio  6503  iotaval  6507  iotanul  6513  iuneq12daf  33033  bj-abv  37652  bj-cleq  37709  abbi1sn  43096  iotain  45244