Theorem sbiedv 2037

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 2038). (Contributed by NM, 7-Jan-2017.)

Hypothesis

Ref	Expression
sbiedv.1	⊢ ((φ ∧ x = y) → (ψ ↔ χ))

Assertion

Ref	Expression
sbiedv	⊢ (φ → ([y / x]ψ ↔ χ))

Distinct variable groups:   φ,x   χ,x

Allowed substitution hints:   φ(y)   ψ(x,y)   χ(y)

Proof of Theorem sbiedv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2		nfvd 1620	. 2 ⊢ (φ → Ⅎxχ)
3		sbiedv.1	. . 3 ⊢ ((φ ∧ x = y) → (ψ ↔ χ))
4	3	ex 423	. 2 ⊢ (φ → (x = y → (ψ ↔ χ)))
5	1, 2, 4	sbied 2036	1 ⊢ (φ → ([y / x]ψ ↔ χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  [wsb 1648

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by: (None)