Theorem equsalhwOLD 1839

Description: Obsolete proof of equsalhw 1838 as of 28-Dec-2017. (Contributed by NM, 29-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
equsalhwOLD	⊢ (∀x(x = y → φ) ↔ ψ)

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem equsalhwOLD

Step	Hyp	Ref	Expression
1		equsalhwOLD.2	. . . . 5 ⊢ (x = y → (φ ↔ ψ))
2		sp 1747	. . . . . 6 ⊢ (∀xψ → ψ)
3		equsalhwOLD.1	. . . . . 6 ⊢ (ψ → ∀xψ)
4	2, 3	impbii 180	. . . . 5 ⊢ (∀xψ ↔ ψ)
5	1, 4	syl6bbr 254	. . . 4 ⊢ (x = y → (φ ↔ ∀xψ))
6	5	pm5.74i 236	. . 3 ⊢ ((x = y → φ) ↔ (x = y → ∀xψ))
7	6	albii 1566	. 2 ⊢ (∀x(x = y → φ) ↔ ∀x(x = y → ∀xψ))
8	3	a1d 22	. . . 4 ⊢ (ψ → (x = y → ∀xψ))
9	3, 8	alrimih 1565	. . 3 ⊢ (ψ → ∀x(x = y → ∀xψ))
10		ax9v 1655	. . . . 5 ⊢ ¬ ∀x ¬ x = y
11		con3 126	. . . . . 6 ⊢ ((x = y → ∀xψ) → (¬ ∀xψ → ¬ x = y))
12	11	al2imi 1561	. . . . 5 ⊢ (∀x(x = y → ∀xψ) → (∀x ¬ ∀xψ → ∀x ¬ x = y))
13	10, 12	mtoi 169	. . . 4 ⊢ (∀x(x = y → ∀xψ) → ¬ ∀x ¬ ∀xψ)
14		ax6o 1750	. . . 4 ⊢ (¬ ∀x ¬ ∀xψ → ψ)
15	13, 14	syl 15	. . 3 ⊢ (∀x(x = y → ∀xψ) → ψ)
16	9, 15	impbii 180	. 2 ⊢ (ψ ↔ ∀x(x = y → ∀xψ))
17	7, 16	bitr4i 243	1 ⊢ (∀x(x = y → φ) ↔ ψ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by: (None)