Theorem sbc7 3074

Description: An equivalence for class substitution in the spirit of df-clab 2340. Note that x and A don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbc7	⊢ ([̣A / x]̣φ ↔ ∃y(y = A ∧ [̣y / x]̣φ))

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

Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem sbc7

Step	Hyp	Ref	Expression
1		sbcco 3069	. 2 ⊢ ([̣A / y]̣[̣y / x]̣φ ↔ [̣A / x]̣φ)
2		sbc5 3071	. 2 ⊢ ([̣A / y]̣[̣y / x]̣φ ↔ ∃y(y = A ∧ [̣y / x]̣φ))
3	1, 2	bitr3i 242	1 ⊢ ([̣A / x]̣φ ↔ ∃y(y = A ∧ [̣y / x]̣φ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642  [̣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-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

This theorem is referenced by: (None)