Theorem sbcbii 3102

Description: Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)

Hypothesis

Ref	Expression
sbcbii.1	⊢ (φ ↔ ψ)

Assertion

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

Proof of Theorem sbcbii

Step	Hyp	Ref	Expression
1		sbcbii.1	. . . 4 ⊢ (φ ↔ ψ)
2	1	a1i 10	. . 3 ⊢ ( ⊤ → (φ ↔ ψ))
3	2	sbcbidv 3101	. 2 ⊢ ( ⊤ → ([̣A / x]̣φ ↔ [̣A / x]̣ψ))
4	3	trud 1323	1 ⊢ ([̣A / x]̣φ ↔ [̣A / x]̣ψ)

Syntax hints:   ↔ wb 176   ⊤ wtru 1316  [̣wsbc 3047

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 3048

This theorem is referenced by:  sbcbiiOLD  3103  eqsbc2  3104  sbccomlem  3117  sbccom  3118  sbcrext  3120  sbcabel  3124  csbco  3146  sbcnel12g  3154  sbcne12g  3155  sbccsbg  3165  sbccsb2g  3166  csbnestgf  3185  csbabg  3198  sbcss  3661  inopab  4863  eqerlem  5961