Theorem sbcbidv 3100

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

Hypothesis

Ref	Expression
sbcbidv.1	⊢ (φ → (ψ ↔ χ))

Assertion

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

Distinct variable group:   φ,x

Allowed substitution hints:   ψ(x)   χ(x)   A(x)

Proof of Theorem sbcbidv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2		sbcbidv.1	. 2 ⊢ (φ → (ψ ↔ χ))
3	1, 2	sbcbid 3099	1 ⊢ (φ → ([̣A / x]̣ψ ↔ [̣A / x]̣χ))

Syntax hints:   → wi 4   ↔ wb 176  [̣wsbc 3046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3047

This theorem is referenced by:  sbcbii  3101  csbcomg  3159  opelopabsb  4697  opelopabf  4711