Theorem infeq123d 8387

Description: Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)

Hypotheses

Ref	Expression
infeq123d.a	⊢ (𝜑 → 𝐴 = 𝐷)
infeq123d.b	⊢ (𝜑 → 𝐵 = 𝐸)
infeq123d.c	⊢ (𝜑 → 𝐶 = 𝐹)

Assertion

Ref	Expression
infeq123d	⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))

Proof of Theorem infeq123d

Step	Hyp	Ref	Expression
1		infeq123d.a	. . 3 ⊢ (𝜑 → 𝐴 = 𝐷)
2		infeq123d.b	. . 3 ⊢ (𝜑 → 𝐵 = 𝐸)
3		infeq123d.c	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐹)
4	3	cnveqd 5298	. . 3 ⊢ (𝜑 → ◡𝐶 = ◡𝐹)
5	1, 2, 4	supeq123d 8356	. 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡𝐶) = sup(𝐷, 𝐸, ◡𝐹))
6		df-inf 8349	. 2 ⊢ inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, ◡𝐶)
7		df-inf 8349	. 2 ⊢ inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, ◡𝐹)
8	5, 6, 7	3eqtr4g 2681	1 ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))

Syntax hints:   → wi 4   = wceq 1483  ^◡ccnv 5113  supcsup 8346  infcinf 8347

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-in 3581  df-ss 3588  df-uni 4437  df-br 4654  df-opab 4713  df-cnv 5122  df-sup 8348  df-inf 8349

This theorem is referenced by:  wsuceq123  31760  wlimeq12  31765