Theorem sbiedh 1787

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1790). New proofs should use sbied 1788 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
sbiedh.1	⊢ (𝜑 → ∀𝑥𝜑)
sbiedh.2	⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
sbiedh.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
sbiedh	⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Proof of Theorem sbiedh

Step	Hyp	Ref	Expression
1		sb1 1766	. . . 4 ⊢ ([𝑦 / 𝑥]𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))
2		sbiedh.1	. . . . 5 ⊢ (𝜑 → ∀𝑥𝜑)
3		sbiedh.3	. . . . . . 7 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
4		biimp 118	. . . . . . 7 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒))
5	3, 4	syl6 33	. . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
6	5	impd 254	. . . . 5 ⊢ (𝜑 → ((𝑥 = 𝑦 ∧ 𝜓) → 𝜒))
7	2, 6	eximdh 1611	. . . 4 ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → ∃𝑥𝜒))
8	1, 7	syl5 32	. . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → ∃𝑥𝜒))
9		sbiedh.2	. . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
10	2, 9	19.9hd 1662	. . 3 ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒))
11	8, 10	syld 45	. 2 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒))
12		biimpr 130	. . . . . . 7 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓))
13	3, 12	syl6 33	. . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜒 → 𝜓)))
14	13	com23 78	. . . . 5 ⊢ (𝜑 → (𝜒 → (𝑥 = 𝑦 → 𝜓)))
15	2, 14	alimdh 1467	. . . 4 ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
16		sb2 1767	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) → [𝑦 / 𝑥]𝜓)
17	15, 16	syl6 33	. . 3 ⊢ (𝜑 → (∀𝑥𝜒 → [𝑦 / 𝑥]𝜓))
18	9, 17	syld 45	. 2 ⊢ (𝜑 → (𝜒 → [𝑦 / 𝑥]𝜓))
19	11, 18	impbid 129	1 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351  ∃wex 1492  [wsb 1762

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-sb 1763

This theorem is referenced by:  sbied  1788  sbieh  1790  sbcomxyyz  1972