Theorem equsalh 1737

Description: A useful equivalence related to substitution. New proofs should use equsal 1738 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem equsalh

Step	Hyp	Ref	Expression
1		equsalh.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2		equsalh.1	. . . . . 6 ⊢ (𝜓 → ∀𝑥𝜓)
3	2	19.3h 1564	. . . . 5 ⊢ (∀𝑥𝜓 ↔ 𝜓)
4	1, 3	bitr4di 198	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))
5	4	pm5.74i 180	. . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → ∀𝑥𝜓))
6	5	albii 1481	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))
7	2	a1d 22	. . . 4 ⊢ (𝜓 → (𝑥 = 𝑦 → ∀𝑥𝜓))
8	2, 7	alrimih 1480	. . 3 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))
9		ax9o 1709	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓) → 𝜓)
10	8, 9	impbii 126	. 2 ⊢ (𝜓 ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))
11	6, 10	bitr4i 187	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sb6x  1790  dvelimfALT2  1828  dvelimALT  2026  dvelimfv  2027  dvelimor  2034