Theorem sbi2v 1886

Description: Reverse direction of sbimv 1887. (Contributed by Jim Kingdon, 18-Jan-2018.)

Assertion

Ref	Expression
sbi2v	⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbi2v

Step	Hyp	Ref	Expression
1		19.38 1670	. . 3 ⊢ ((∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)) → ∀𝑥((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜓)))
2		pm3.3 259	. . . . 5 ⊢ (((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜓)) → (𝑥 = 𝑦 → (𝜑 → (𝑥 = 𝑦 → 𝜓))))
3		pm2.04 82	. . . . 5 ⊢ ((𝜑 → (𝑥 = 𝑦 → 𝜓)) → (𝑥 = 𝑦 → (𝜑 → 𝜓)))
4	2, 3	syli 37	. . . 4 ⊢ (((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜓)) → (𝑥 = 𝑦 → (𝜑 → 𝜓)))
5	4	alimi 1449	. . 3 ⊢ (∀𝑥((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜓)) → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))
6	1, 5	syl 14	. 2 ⊢ ((∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)) → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))
7		sb5 1881	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
8		sb6 1880	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
9	7, 8	imbi12i 238	. 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
10		sb6 1880	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))
11	6, 9, 10	3imtr4i 200	1 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1347  ∃wex 1486  [wsb 1756

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 1441  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-11 1500  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529

This theorem depends on definitions:  df-bi 116  df-sb 1757

This theorem is referenced by:  sbimv  1887