Theorem equsalhw 1838

Description: Weaker version of equsalh 1961 (requiring distinct variables) without using ax-12 1925. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 28-Dec-2017.)

Hypotheses

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

Assertion

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

Distinct variable group:   x,y

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

Proof of Theorem equsalhw

Step	Hyp	Ref	Expression
1		equsalhw.1	. . 3 ⊢ (ψ → ∀xψ)
2	1	19.23h 1802	. 2 ⊢ (∀x(x = y → ψ) ↔ (∃x x = y → ψ))
3		equsalhw.2	. . . 4 ⊢ (x = y → (φ ↔ ψ))
4	3	pm5.74i 236	. . 3 ⊢ ((x = y → φ) ↔ (x = y → ψ))
5	4	albii 1566	. 2 ⊢ (∀x(x = y → φ) ↔ ∀x(x = y → ψ))
6		a9ev 1656	. . 3 ⊢ ∃x x = y
7	6	a1bi 327	. 2 ⊢ (ψ ↔ (∃x x = y → ψ))
8	2, 5, 7	3bitr4i 268	1 ⊢ (∀x(x = y → φ) ↔ ψ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541

This proof depends on 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 proof depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is used by:  dvelimhw  1849