Theorem eleq2dALT 2826

Description: Alternate proof of eleq2d 2825, shorter at the expense of requiring ax-12 2189. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 20-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
eleq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
eleq2dALT	⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))

Proof of Theorem eleq2dALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1d.1	. . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐵)
2		dfcleq 2732	. . . . . 6 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	1, 2	sylib 219	. . . . 5 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
4	3	19.21bi 2201	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	anbi2d 636	. . 3 ⊢ (𝜑 → ((𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵)))
6	5	exbidv 1928	. 2 ⊢ (𝜑 → (∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵)))
7		dfclel 2815	. 2 ⊢ (𝐶 ∈ 𝐴 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐴))
8		dfclel 2815	. 2 ⊢ (𝐶 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐶 ∧ 𝑥 ∈ 𝐵))
9	6, 7, 8	3bitr4g 315	1 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2731  df-clel 2814

This theorem is referenced by: (None)