Theorem sbhypf 1942

Description: Introduce an explicit substitution into an implicit substitution hypothesis.

Hypotheses

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

Assertion

Ref	Expression
sbhypf	⊢ (y = A → ([y / x]φ ↔ ψ))

Distinct variable groups:   x,A   x,y

Proof of Theorem sbhypf

Step	Hyp	Ref	Expression
1		visset 1816	. . 3 ⊢ y ∈ V
2		eqeq1 1484	. . 3 ⊢ (x = y → (x = A ↔ y = A))
3	1, 2	ceqsexv 1838	. 2 ⊢ (∃x(x = y ⋀ x = A) ↔ y = A)
4		hbs1 1334	. . . 4 ⊢ ([y / x]φ → ∀x[y / x]φ)
5		sbhypf.1	. . . 4 ⊢ (ψ → ∀xψ)
6	4, 5	hbbi 1012	. . 3 ⊢ (([y / x]φ ↔ ψ) → ∀x([y / x]φ ↔ ψ))
7		sbequ12 1183	. . . . 5 ⊢ (x = y → (φ ↔ [y / x]φ))
8	7	bicomd 523	. . . 4 ⊢ (x = y → ([y / x]φ ↔ φ))
9		sbhypf.2	. . . 4 ⊢ (x = A → (φ ↔ ψ))
10	8, 9	sylan9bb 542	. . 3 ⊢ ((x = y ⋀ x = A) → ([y / x]φ ↔ ψ))
11	6, 10	19.23ai 1066	. 2 ⊢ (∃x(x = y ⋀ x = A) → ([y / x]φ ↔ ψ))
12	3, 11	sylbir 201	1 ⊢ (y = A → ([y / x]φ ↔ ψ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956   = wceq 958  ∃wex 982  [wsbc 1172

This theorem is referenced by:  ac6sf 4770

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-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-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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