Theorem sbcieg 1964

Description: Conversion of implicit substitution to explicit class substitution.

Hypothesis

Ref	Expression
sbcieg.1	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
sbcieg	⊢ (A ∈ B → ([A / x]φ ↔ ψ))

Distinct variable groups:   x,A   ψ,x

Proof of Theorem sbcieg

Step	Hyp	Ref	Expression
1		elisset 1820	. 2 ⊢ (A ∈ B → A ∈ V)
2		ax-17 973	. . . 4 ⊢ (ψ → ∀xψ)
3	2	a1i 8	. . 3 ⊢ (A ∈ V → (ψ → ∀xψ))
4		sbcieg.1	. . 3 ⊢ (x = A → (φ ↔ ψ))
5	3, 4	sbciegf 1963	. 2 ⊢ (A ∈ V → ([A / x]φ ↔ ψ))
6	1, 5	syl 10	1 ⊢ (A ∈ B → ([A / x]φ ↔ ψ))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 956   = wceq 958   ∈ wcel 960  [wsbc 1172  Vcvv 1814

This theorem is referenced by:  sbcie 1965

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945

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