Theorem sbcco3gOLD 3193

Description: Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sbcco3g.1	⊢ (x = A → B = C)

Assertion

Ref	Expression
sbcco3gOLD	⊢ ((A ∈ V ∧ ∀x B ∈ W) → ([̣A / x]̣[̣B / y]̣φ ↔ [̣C / y]̣φ))

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

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

Proof of Theorem sbcco3gOLD

Step	Hyp	Ref	Expression
1		sbcco3g.1	. . 3 ⊢ (x = A → B = C)
2	1	sbcco3g 3192	. 2 ⊢ (A ∈ V → ([̣A / x]̣[̣B / y]̣φ ↔ [̣C / y]̣φ))
3	2	adantr 451	1 ⊢ ((A ∈ V ∧ ∀x B ∈ W) → ([̣A / x]̣[̣B / y]̣φ ↔ [̣C / y]̣φ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  [̣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-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)