Theorem infeq123d 7132

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 4861	. . 3 ⊢ (𝜑 → ◡𝐶 = ◡𝐹)
5	1, 2, 4	supeq123d 7107	. 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡𝐶) = sup(𝐷, 𝐸, ◡𝐹))
6		df-inf 7101	. 2 ⊢ inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, ◡𝐶)
7		df-inf 7101	. 2 ⊢ inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, ◡𝐹)
8	5, 6, 7	3eqtr4g 2264	1 ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))

Syntax hints:   → wi 4   = wceq 1373  ^◡ccnv 4681  supcsup 7098  infcinf 7099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-in 3176  df-ss 3183  df-uni 3856  df-br 4051  df-opab 4113  df-cnv 4690  df-sup 7100  df-inf 7101

This theorem is referenced by: (None)