Theorem sbieh 1783

Description: Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1784 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
sbieh	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Proof of Theorem sbieh

Step	Hyp	Ref	Expression
1		id 19	. 2 ⊢ (𝜑 → 𝜑)
2	1	hbth 1456	. . 3 ⊢ ((𝜑 → 𝜑) → ∀𝑥(𝜑 → 𝜑))
3		sbieh.1	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	a1i 9	. . 3 ⊢ ((𝜑 → 𝜑) → (𝜓 → ∀𝑥𝜓))
5		sbieh.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	5	a1i 9	. . 3 ⊢ ((𝜑 → 𝜑) → (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)))
7	2, 4, 6	sbiedh 1780	. 2 ⊢ ((𝜑 → 𝜑) → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
8	1, 7	ax-mp 5	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-sb 1756

This theorem is referenced by:  sbie  1784  sbco2vlem  1937  equsb3lem  1943  sbco2yz  1956  dvelimf  2008  elsb1  2148  elsb2  2149