Theorem sbcbidv 3009

Description: Formula-building deduction 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 1516	. 2 ⊢ Ⅎ𝑥𝜑
2		sbcbidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	sbcbid 3008	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒))

Syntax hints:   → wi 4   ↔ wb 104  [wsbc 2951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-sbc 2952

This theorem is referenced by:  sbcbii  3010  csbcomg  3068  opelopabsb  4238  opelopabf  4252  sbcfng  5335  sbcfg  5336  f1od2  6203