Theorem sbcimdv 3108

Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)

Hypothesis

Ref	Expression
sbcimdv.1	⊢ (φ → (ψ → χ))

Assertion

Ref	Expression
sbcimdv	⊢ ((φ ∧ A ∈ V) → ([̣A / x]̣ψ → [̣A / x]̣χ))

Distinct variable group:   φ,x

Allowed substitution hints:   ψ(x)   χ(x)   A(x)   V(x)

Proof of Theorem sbcimdv

Step	Hyp	Ref	Expression
1		sbcimdv.1	. . . . 5 ⊢ (φ → (ψ → χ))
2	1	alrimiv 1631	. . . 4 ⊢ (φ → ∀x(ψ → χ))
3		spsbc 3059	. . . 4 ⊢ (A ∈ V → (∀x(ψ → χ) → [̣A / x]̣(ψ → χ)))
4	2, 3	syl5 28	. . 3 ⊢ (A ∈ V → (φ → [̣A / x]̣(ψ → χ)))
5		sbcimg 3088	. . 3 ⊢ (A ∈ V → ([̣A / x]̣(ψ → χ) ↔ ([̣A / x]̣ψ → [̣A / x]̣χ)))
6	4, 5	sylibd 205	. 2 ⊢ (A ∈ V → (φ → ([̣A / x]̣ψ → [̣A / x]̣χ)))
7	6	impcom 419	1 ⊢ ((φ ∧ A ∈ V) → ([̣A / x]̣ψ → [̣A / x]̣χ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  [̣wsbc 3047

This proof depends on 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 proof 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 used by: (None)