Theorem equsalhw 2123

Description: Weaker version of equsalh 2294 with a dv condition which does not require ax-13 2246. (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 2122	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓))
3		equsalhw.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	pm5.74i 260	. . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓))
5	4	albii 1747	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
6		ax6ev 1890	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
7	6	a1bi 352	. 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓))
8	2, 5, 7	3bitr4i 292	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by:  dvelimhw  2173