Theorem sbcco2 3070

Description: A composition law for class substitution. Importantly, x may occur free in the class expression substituted for A. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypothesis

Ref	Expression
sbcco2.1	⊢ (x = y → A = B)

Assertion

Ref	Expression
sbcco2	⊢ ([̣x / y]̣[̣B / x]̣φ ↔ [̣A / x]̣φ)

Distinct variable groups:   x,y   φ,y   y,A

Allowed substitution hints:   φ(x)   A(x)   B(x,y)

Proof of Theorem sbcco2

Step	Hyp	Ref	Expression
1		sbsbc 3051	. 2 ⊢ ([x / y][̣B / x]̣φ ↔ [̣x / y]̣[̣B / x]̣φ)
2		nfv 1619	. . 3 ⊢ Ⅎy[̣A / x]̣φ
3		sbcco2.1	. . . . 5 ⊢ (x = y → A = B)
4	3	eqcoms 2356	. . . 4 ⊢ (y = x → A = B)
5		dfsbcq 3049	. . . . 5 ⊢ (A = B → ([̣A / x]̣φ ↔ [̣B / x]̣φ))
6	5	bicomd 192	. . . 4 ⊢ (A = B → ([̣B / x]̣φ ↔ [̣A / x]̣φ))
7	4, 6	syl 15	. . 3 ⊢ (y = x → ([̣B / x]̣φ ↔ [̣A / x]̣φ))
8	2, 7	sbie 2038	. 2 ⊢ ([x / y][̣B / x]̣φ ↔ [̣A / x]̣φ)
9	1, 8	bitr3i 242	1 ⊢ ([̣x / y]̣[̣B / x]̣φ ↔ [̣A / x]̣φ)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648  [̣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: (None)