Theorem sbciegf 1956

Description: Conversion of implicit substitution to explicit class substitution.

Hypotheses

Ref	Expression
sbciegf.1	⊢ (A ∈ B → (ψ → ∀xψ))
sbciegf.2	⊢ (x = A → (φ ↔ ψ))

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem sbciegf

Step	Hyp	Ref	Expression
1		sbciegf.1	. . 3 ⊢ (A ∈ B → (ψ → ∀xψ))
2	1	19.21aiv 1284	. 2 ⊢ (A ∈ B → ∀x(ψ → ∀xψ))
3		sbciegf.2	. . . 4 ⊢ (x = A → (φ ↔ ψ))
4	3	ax-gen 961	. . 3 ⊢ ∀x(x = A → (φ ↔ ψ))
5		sbciegft 1955	. . . 4 ⊢ ((A ∈ B ⋀ ∀x(ψ → ∀xψ) ⋀ ∀x(x = A → (φ ↔ ψ))) → ([A / x]φ ↔ ψ))
6	5	3exp 831	. . 3 ⊢ (A ∈ B → (∀x(ψ → ∀xψ) → (∀x(x = A → (φ ↔ ψ)) → ([A / x]φ ↔ ψ))))
7	4, 6	mpii 45	. 2 ⊢ (A ∈ B → (∀x(ψ → ∀xψ) → ([A / x]φ ↔ ψ)))
8	2, 7	mpd 26	1 ⊢ (A ∈ B → ([A / x]φ ↔ ψ))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 952   = wceq 954   ∈ wcel 956  [wsbc 1168

This theorem is referenced by:  sbcieg 1957  sbcbrg 2657

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-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 776  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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