Theorem sbcbid 3089

Description: Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)

Hypotheses

Ref	Expression
sbcbid.1	⊢ Ⅎ𝑥𝜑
sbcbid.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
sbcbid	⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒))

Proof of Theorem sbcbid

Step	Hyp	Ref	Expression
1		sbcbid.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		sbcbid.2	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	abbid 2348	. . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
4	3	eleq2d 2301	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐴 ∈ {𝑥 ∣ 𝜒}))
5		df-sbc 3032	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
6		df-sbc 3032	. 2 ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝐴 ∈ {𝑥 ∣ 𝜒})
7	4, 5, 6	3bitr4g 223	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒))

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1508   ∈ wcel 2202  {cab 2217  [wsbc 3031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-sbc 3032

This theorem is referenced by:  sbcbidv  3090  csbeq2d  3152  bezoutlemstep  12567