Theorem vsbcint 1438

Description: Change variable of an implicit substitution hypothesis, introducing an explicit substitution. (Contributed by Raph Levien, 10-Apr-04.)

Hypothesis

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

Assertion

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

Distinct variable group(s):   ψ,x,y   x,A

Proof of Theorem vsbcint

Step	Hyp	Ref	Expression
1		visset 1350	. . 3 ⊢ y ∈ V
2		cleq1 1107	. . 3 ⊢ (x = y → (x = A ↔ y = A))
3	1, 2	ceqsexv 1371	. 2 ⊢ (∃x(x = y ∧ x = A) ↔ y = A)
4		hbs1 986	. . . 4 ⊢ ([y / x]φ → ∀x[y / x]φ)
5		ax-17 925	. . . 4 ⊢ (ψ → ∀xψ)
6	4, 5	hbbi 705	. . 3 ⊢ (([y / x]φ ↔ ψ) → ∀x([y / x]φ ↔ ψ))
7		sbequ12 865	. . . . 5 ⊢ (x = y → (φ ↔ [y / x]φ))
8	7	bicomd 399	. . . 4 ⊢ (x = y → ([y / x]φ ↔ φ))
9		vsbcint.1	. . . 4 ⊢ (x = A → (φ ↔ ψ))
10	8, 9	sylan9bb 418	. . 3 ⊢ ((x = y ∧ x = A) → ([y / x]φ ↔ ψ))
11	6, 10	19.23ai 746	. 2 ⊢ (∃x(x = y ∧ x = A) → ([y / x]φ ↔ ψ))
12	3, 11	sylbir 176	1 ⊢ (y = A → ([y / x]φ ↔ ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797  [wsb 852   = wceq 1091

This theorem is referenced by:  nn0ind 4612

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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