Theorem sbceqbii 36235

Description: Formula-building inference for class substitution. General version of sbcbii 3793. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
sbceqbii.1	⊢ 𝐴 = 𝐵
sbceqbii.2	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
sbceqbii	⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜓)

Proof of Theorem sbceqbii

Step	Hyp	Ref	Expression
1		sbceqbii.1	. . 3 ⊢ 𝐴 = 𝐵
2		sbceqbii.2	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	abbii 2798	. . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}
4	1, 3	eleq12i 2824	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐵 ∈ {𝑥 ∣ 𝜓})
5		df-sbc 3737	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
6		df-sbc 3737	. 2 ⊢ ([𝐵 / 𝑥]𝜓 ↔ 𝐵 ∈ {𝑥 ∣ 𝜓})
7	4, 5, 6	3bitr4i 303	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜓)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2111  {cab 2709  [wsbc 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-sbc 3737

This theorem is referenced by: (None)