Theorem drnfc2 2960

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)

Hypothesis

Ref	Expression
drnfc1.1	⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
drnfc2	⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵))

Proof of Theorem drnfc2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		drnfc1.1	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)
2	1	eleq2d 2864	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
3	2	drnf2 2451	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧 𝑤 ∈ 𝐴 ↔ Ⅎ𝑧 𝑤 ∈ 𝐵))
4	3	dral2 2445	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐴 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐵))
5		df-nfc 2930	. 2 ⊢ (Ⅎ𝑧𝐴 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐴)
6		df-nfc 2930	. 2 ⊢ (Ⅎ𝑧𝐵 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐵)
7	4, 5, 6	3bitr4g 306	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1651   = wceq 1653  Ⅎwnf 1879   ∈ wcel 2157  Ⅎwnfc 2928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-cleq 2792  df-clel 2795  df-nfc 2930

This theorem is referenced by: (None)