Theorem sbc19.20dv 1981

Description: Substitution analog of Theorem 19.20 of [Margaris] p. 90.

Hypothesis

Ref	Expression
sbc19.20dv.1	⊢ (φ → (ψ → χ))

Assertion

Ref	Expression
sbc19.20dv	⊢ ((φ ⋀ A ∈ B) → ([A / x]ψ → [A / x]χ))

Distinct variable group:   φ,x

Proof of Theorem sbc19.20dv

Step	Hyp	Ref	Expression
1		a4sbc 1941	. . . 4 ⊢ (A ∈ B → (∀x(ψ → χ) → [A / x](ψ → χ)))
2		sbc19.20dv.1	. . . . 5 ⊢ (φ → (ψ → χ))
3	2	19.21aiv 1284	. . . 4 ⊢ (φ → ∀x(ψ → χ))
4	1, 3	syl5 21	. . 3 ⊢ (A ∈ B → (φ → [A / x](ψ → χ)))
5		sbcimg 1966	. . 3 ⊢ (A ∈ B → ([A / x](ψ → χ) ↔ ([A / x]ψ → [A / x]χ)))
6	4, 5	sylibd 202	. 2 ⊢ (A ∈ B → (φ → ([A / x]ψ → [A / x]χ)))
7	6	impcom 351	1 ⊢ ((φ ⋀ A ∈ B) → ([A / x]ψ → [A / x]χ))

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 952   ∈ wcel 956  [wsbc 1168

This theorem is referenced by:  fsum1s 6955  fsump1s 6959

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938

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