Theorem bj-equsexval 32638

Description: Special case of equsexv 2109 proved from Tarski, ax-10 2019 (modal5) and hba1 2151 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-equsexval.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))

Assertion

Ref	Expression
bj-equsexval	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bj-equsexval

Step	Hyp	Ref	Expression
1		bj-equsexval.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓))
2	1	pm5.32i 669	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝑦 ∧ ∀𝑥𝜓))
3	2	exbii 1774	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓))
4		ax6ev 1890	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
5		bj-19.41al 32637	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓) ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝜓))
6	4, 5	mpbiran 953	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑥𝜓) ↔ ∀𝑥𝜓)
7	3, 6	bitri 264	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀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-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)