Theorem sbcbid 2810

Description: Formula-building deduction rule 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 2151	. . 3 ⊢ (φ → {x ∣ ψ} = {x ∣ χ})
4	3	eleq2d 2104	. 2 ⊢ (φ → (A ∈ {x ∣ ψ} ↔ A ∈ {x ∣ χ}))
5		df-sbc 2759	. 2 ⊢ ([A / x]ψ ↔ A ∈ {x ∣ ψ})
6		df-sbc 2759	. 2 ⊢ ([A / x]χ ↔ A ∈ {x ∣ χ})
7	4, 5, 6	3bitr4g 212	1 ⊢ (φ → ([A / x]ψ ↔ [A / x]χ))

Syntax hints:   → wi 4   ↔ wb 98  Ⅎwnf 1346   ∈ wcel 1390  {cab 2023  [wsbc 2758

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-sbc 2759

This theorem is referenced by:  sbcbidv  2811  csbeq2d  2868